Heterogeneous fibring of deductive systems

via abstract proof systems

Luıs Cruz-Filipe Amılcar Sernadas Cristina SernadasSQIG - IT and CLC - IST, Portugallcf,acs,css@math.ist.utl.pt

Abstract

Fibring is a meta-logical constructor that applied to two logics producesa new logic whose formulas allow the mixing of symbols. Homogeneousfibring assumes that the original logics are presented in the same way (e.gvia Hilbert calculi). Heterogeneous fibring, allowing the original logicsto have different presentations (e.g. one presented by a Hilbert calculusand the other by a sequent calculus), has been an open problem. Herein,consequence systems are shown to be a good solution for heterogeneousfibring when one of the logics is presented in a semantic way and the otherby a calculus and also a solution for the heterogeneous fibring of calculi.The new notion of abstract proof system is shown to provide a bettersolution to heterogeneous fibring of calculi namely because derivations inthe fibring keep the constructive nature of derivations in the original logics.Preservation of compactness and semi-decidability is investigated.Keywords: heterogeneous fibring, abstract proof system, preservation.AMS Classification: 03B22, 03F07

1 Introduction

A mechanism for combining logics is an operation on a (sub)class of logics inthe sense that it provides the means for obtaining a new logic from a finitenumber of logics (for instance, fusion is an operation on the subclass of modallogics while fibring is an operation on the class of logics). For a nice and gentlemotivation of the topic see [3], and for an early example of the combinationof tense and modality see [24]. The different methods for combination dependupon and impose different presentations of the original logics ranging, on onehand, from very abstract to more concrete ones and, on the other hand, fromdeductive-based to semantic-based. In general, it is assumed that the logicsto be combined are presented in the same way. For instance, the logics to becombined are endowed with a Hilbert calculus.

In this paper, we address the problem of combining logics presented indifferent styles (e.g. a Hilbert calculus and a sequent calculus). This is an openproblem of great practical interest, since the requirement that the two logics tobe combined be presented in the same way is often not met in practice.

A very abstract presentation of a logic is via consequence systems (with nodeductive or semantic connotation). A consequence system is a pair composed

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by a set (of formulas, usually with no details on the construction) and a binary(consequence) relation (the pairs 〈Γ, ϕ〉 indicate that formula ϕ is a consequenceof set of formulas Γ). Some properties are required for this consequence relation.The presentation of logics via consequence systems allows the definition of somesimple forms of combining logics like, for example, union of logics (the set offormulas in the union is the union of the sets of formulas of the components andthe consequence relation is also the union of the original consequence relations).

More interesting combination mechanisms can be defined when describinglogics in a more concrete way. In the case of fibring (see [11, 12]), it is necessaryto indicate the signatures (set of symbols) of the (usually two) original logics.The set of formulas is the free algebra generated by the set of symbols in thesignature and a set of (schema) variables. A signature of the fibring is theunion of original signatures (but the set of formulas is not the union of the setof formulas of the components since in the same formula we can have symbolsfrom both signatures). Signatures are also needed for the fusion of modal logics,see [24], as well as when combining temporal logic systems, see [9].

Also the consequence relation can be more concrete, for instance, when itis generated from a Hilbert calculus (with axioms and rules). The inducedconsequence relation is then defined in a constructive way. Another possibilityfor defining the consequence system is via semantics by giving pairs composedby a class of models and a satisfaction (binary) relation (a pair 〈m, ϕ〉 meansthat model m satisfies formula ϕ). In the induced consequence system, theconsequence relation is then the semantic entailment.

Fusion of two normal modal logics presented by Hilbert calculi is a bi-modallogic presented by an Hilbert calculus whose axioms and rules are the union ofthe axioms and rules for both of them. But fusion of modal logics can also beexplained in semantic terms. The models of the fusion are bi-Kripke structureswith the same set of worlds but different accessibility relations. Similar exam-ples can be given for fibring. For example, the fibring of logics presented byHilbert calculi is a logic presented by a Hilbert calculus having the axioms andthe rules of the component Hilbert calculi. Also some results on the fibring oftableau systems can be found in [8, 1].

In the examples above, we are implicitly thinking of combining logics ina homogeneous scenario: both logics are presented in the same way eitherby Hilbert calculi or by Kripke structures. However, this is not usually thecase. Heterogeneous combination of logics, in particular heterogeneous fibring,is an open problem identified by Gabbay in [11]. That is, we would like tobe able to combine logics presented in a different way, for instance, to definethe combination of two modal logics, one deductively presented by a Hilbertcalculus and the other semantically presented by Kripke structures. Note thatheterogeneous combination was never dealt with any of the existing combinationmechanisms.

The heterogeneous scenario was dealt with in [21] for fibring logics pre-sented semantically but with different semantic domains, for instance fibringmodal logic presented by Kripke structures with first-order logic with the usualfirst-order structures. The solution was to introduce an algebraic structure asthe main semantic primitive and to define fibring of such algebraic structures.

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Hence when we are given a logic presented semantically, the first step is todetermine how to extract an algebraic structure from each model. A criterionfor the correction of the extraction mechanism is to prove that the entailmentof the original logics is preserved.

However, so far, there was no solution for the problems of combining twologics when: (i) one is presented in a semantic way and the other is presentedby a calculus (Hilbert, sequent, tableau, etc); (ii) both are presented by calculibut these are of different kinds, say one Hilbert and the other sequent. Thissituation may arise in practice: an example would be the study of the behaviorthrough time of a system whose state logic is presented via a Hilbert calculusand where the temporal logic is given as a sequent calculus. Applying thecurrent-day fibring techniques would require changing the presentation of one(or both) of the logics, which is neither easy nor convenient. This is the problemwe address in this paper.

A very important issue in combination of logics is preservation of properties.That is, assuming that the original logics have a given property (say decidabil-ity), we want to know whether the combination still has this property (of beingdecidable). In several cases, preservation holds provided that we restrict thelogics at hand: that is, only when we work in a particular subclass of the classof logics.

Among the methods for combining logics, fusion of modalities (logics pre-sented by Hilbert calculi at the deductive level and Kripke structures at thesemantic level) [24] is the best understood in what concerns preservation ofproperties as soundness, weak completeness, uniform Craig interpolation (fortheoremhood) and decidability via finite model property (see [26, 19, 13]). Fur-ther results on preservation of weak completeness can be seen in [10]. Preser-vation results were also obtained in the context of temporalization (addinga temporal dimension to an original logic) as in [27, 18]. Some interestingpreservation properties can also be found for the product of (modal) logics,see [13, 14, 15, 16].

Fibring is a more general mechanism since it goes beyond modal logic. Forinstance, we can think of fibring relevance logic with intuitionistic logic, ormodal logic and first-order logic. Although preservation of soundness, com-pleteness and interpolation has been already investigated in the context ofpropositional-based logics [28, 23, 5], first-order quantification [22], higher-orderquantification [7], non truth-functional semantics [4], sequent calculus and otherdeductive systems [17, 20], other forms of preservation are still to be fully un-derstood, namely the ones related to compactness, decidability and complexity.

The main objective of this paper is to provide solutions to open problems inheterogeneous fibring. The solution to (i), which is also a solution to (ii), is todefine fibring of consequence systems (observe that our notion of consequencesystem differs slightly from the usual one because we need to include the signa-ture as a component). However, the constructive nature of derivations is lost.Hence we provide another solution to (ii) introducing the new concept of ab-stract proof system. We define fibring of proof systems and keep, in the fibring,the constructive nature of derivations. Examples are provided for modal logic.Preservation of compactness, semi-decidability and effectiveness is investigated.

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The paper is organized as follows. Section 2 concentrates on consequencesystems as a possible solution to heterogeneous fibring. Fibring of consequencesystems is defined as a (Tarski) fixed point operator. Section 3 focuses on thenew notion of (abstract) proof system. We introduce the proof systems inducedby Hilbert, sequent and tableau calculi and define fibring of proof systems. Inboth settings several preservation results are proved. Section 4 introduces someauxiliary notions for those not so familiar with computability issues.

2 Consequence systems

We start by defining consequence system and identifying several classes of conse-quence systems (e.g. closed for substitution, compact or finitary and (strongly)semi-decidable). We prove some results about the relationship between theseclasses. Then we show how different kinds of calculi generate consequence sys-tems. The last subsection is dedicated to fibring consequence systems.

When discussing decidability results, we will rely heavily on the Church–Markov–Turing postulate and work with the intuitive concepts.

2.1 Basics

We are interested in dealing with propositional-based logics in order to investi-gate the issue of heterogeneous fibring in a simple context.

A signature C is a family of sets indexed by the natural numbers. Theelements of each Ck are called constructors or connectives of arity k. We saythat C ⊆ C ′ if Ck ⊆ C ′

k for every k ∈ N.Let L(C, Ξ) be the free algebra over C generated by Ξ = ξn : n ∈ N.

In the sequel we will write L(C) instead of L(C, Ξ). The elements of L(C)are called formulas and L(C) is the language. The elements of Ξ are schemavariables that will allow the definition of schematic derivations. A derivationcan be obtained from a schematic derivation by using a substitution.

A substitution is any map σ : Ξ → L(C). Substitutions can be inductivelyextended to formulas and to sets of formulas: σ(γ) is the formula where eachξ ∈ Ξ is replaced by σ(ξ); σ(Γ) = σ(γ) : γ ∈ Γ.

A consequence system is a tuple 〈C,`〉 where C is a signature and `: ℘L →℘L is a map with the following properties:

• Extensivity: Γ ⊆ Γ`;

• Monotonicity: If Γ1 ⊆ Γ2 then Γ`1 ⊆ Γ`2 ;

• Idempotence: (Γ`)` ⊆ Γ`;

• Closure for renaming substitutions: ρ(Γ`) ⊆ (ρ(Γ))` for every renamingsubstitution ρ (that is, a map such that ρ(ξ) ∈ Ξ for every ξ ∈ Ξ).

We say that Γ` is the closure of Γ. Note that (Γ`)` = Γ` is a trivial consequenceof the properties above.

A pair 〈C,`〉 is a quasi consequence system if ` has all the above proper-ties with the possible exception of idempotence. A very simple though not so

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interesting case is the one where Γ` = Γ for every Γ ⊆ L(C). Such a con-sequence system is a topological (Kuratowski) closure operator since it alsosatisfies ∅` = ∅ and (Γ1 ∪ Γ2)` = Γ`1 ∪ Γ`2 for every Γ1, Γ2 ⊆ L(C). But the re-ally interesting consequence systems do not satisfy empty and union propertiesalthough they can enjoy other properties.

A consequence system is said to be non-trivial if ξ /∈ Π` for every ξ ∈ Ξ \Πand Π ⊆ Ξ. A consequence system is closed for substitution if σ(Γ`) ⊆ (σ(Γ))`

for any substitution σ. A consequence system is compact or finitary if Γ` =⋃Φ∈℘finΓ Φ` for every Γ ⊆ L(C), where ℘finΓ is the set of all finite subsets of Γ.

A consequence system 〈C,`〉 is semi-decidable if Γ` is recursively enu-merable1 whenever Γ is recursive. A consequence system is strongly semi-decidable if Γ` is a recursively enumerable set for every recursively enumerableset Γ ⊆ L(C). A natural question is whether there is a consequence systemwhere the closure of a recursive set is not always a recursively enumerable set.We provide the following illustration.

Example 2.1 Consider the consequence system 〈N,`〉 such that Γ` = Γ ∪ Awhere A is any non recursively enumerable set. Then ∅` is not a recursivelyenumerable set even though ∅ is recursive. /

The relationship between semi-decidable and strongly semi-decidable conse-quence systems can be investigated. We observe that a strongly semi-decidableconsequence system is semi-decidable, which is a consequence of the fact thatevery recursive set is a recursively enumerable set.

Proposition 2.2 A compact and semi-decidable consequence system is stronglysemi-decidable.

Proof: Let C be a compact and semi-decidable consequence systemand Γ ⊆ L(C) be a recursively enumerable set. Then either Γ is the empty

set or there is a total recursive function f : N → L(C) such that f(N) = Γ. IfΓ = ∅ then Γ is recursive, and so by hypothesis Γ` is recursively enumerable.In the other case consider an enumeration Γn : n ∈ N of ℘finΓ. (For example,given n ∈ N, write n as a sum of powers of two

n =blog2(n)c∑

i=0

ai2i

and let Γn = f(i) : ai = 1. If Φ = ϕ0, . . . , ϕm ⊆ Γ, then there are numbersi0, . . . , im such that f(ik) = ϕk, and then

Γ∑mk=0 2ik = Φ.)

For all n, Γn is recursive (since it is finite), so Γn is a recursively enumerableset. Therefore, ⋃

n∈NΓ`n

1A set X is recursive if there is an algorithm that decides whether x ∈ X or not; a set Xis recursively enumerable if there is a procedure that answers affirmatively to the question ofwhether x ∈ X whenever that is the case, but may never answer otherwise.

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is a recursively enumerable set (it is a recursive union of recursively enumerablesets), and since C is compact this set coincides with Γ`. ¦

Consequence systems can be related. We say that consequence system 〈C,`〉is weaker than consequence system 〈C ′,`′〉, indicated by

〈C,`〉 ≤ 〈C ′,`′〉,

if C ⊆ C ′ and Γ` ⊆ Γ`′ for every Γ ⊆ L(C). The relation ≤ introduced in theclass of consequence systems is reflexive, transitive and anti-symmetric.

2.2 Induced consequence systems

We now show that different kinds of (general) calculi, as well as logics presentedvia their semantics, induce consequence systems. We stress that the notion ofconsequence system also covers calculi having infinitary rules like some versionsof linear temporal logic where the rule

Xnξ : n ∈ NGξ

is included. This rule means that if ξ holds now (X0ξ) and in each subsequentinstant (Xnξ) then it always holds in the future (Gξ). Later on we discuss anon-compact situation.

2.2.1 Hilbert calculi

A Hilbert calculus is a pair H = 〈C, R〉 where C is a signature and R is a setof (Hilbert) rules, i.e. pairs 〈Θ, η〉 with Θ∪ η ⊆ L(C) and Θ finite. Rules areschematic in the sense that the elements of Ξ can be instantiated. A rule whereΘ = ∅ is said to be an axiom.

The formula ϕ is Hilbert-derived from the set of formulas Γ, indicated byΓ `H ϕ, iff there is a finite sequence (a derivation) ϕ1 . . . ϕn of formulas suchthat ϕn is ϕ and for each i = 1, . . . , n one of the following holds.

• ϕi is an element of Γ (justified by Hyp);

• there exist a rule r = 〈Θ, η〉 and a substitution σ such that ϕi = σ(η) andσ(Θ) ⊆ ϕ1, . . . , ϕi−1 (justified by r).

The following result is easy to prove observing that the pair 〈σ(Θ), σ(η)〉 isan instance of the rule 〈Θ, η〉.

Proposition 2.3 A Hilbert calculus H induces a consequence system C(H) =〈C,`H〉 such that Γ`H = ϕ : Γ `H ϕ. Furthermore, C(H) is compact andclosed for substitution.

Example 2.4 A Hilbert calculus HB = 〈C, R〉 for modal logic with the Baxiom (sound with respect to symmetric frames) is as follows:

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• C0 = Π, C1 = ¬, ¤, C2 = ⇒;• R = 〈∅, (ξ1 ⇒ (ξ2 ⇒ ξ1))〉,

〈∅, ((ξ1 ⇒ (ξ2 ⇒ ξ3))⇒ ((ξ1 ⇒ ξ2)⇒ (ξ1 ⇒ ξ3)))〉,〈∅, (((¬ ξ1)⇒ (¬ ξ2))⇒ (ξ2 ⇒ ξ1))〉,〈∅, ((¤(ξ1 ⇒ ξ2))⇒ ((¤ξ1)⇒ (¤ξ2)))〉,〈ξ1, (ξ1 ⇒ ξ2), ξ2〉,〈ξ1, (¤ξ1)〉〈∅, (ξ1 ⇒ (¤(♦(ξ1))))〉.

Observe that the rules are schematic: the elements of Ξ can be instantiated byany formulas. The last rule is usually known as the B axiom. All the otherrules are the usual ones for normal modal logic K. /

2.2.2 Sequent calculi

A sequent over a signature C is a pair 〈∆1, ∆2〉, denoted by ∆1 → ∆2, where ∆1

(the antecedent) and ∆2 (the consequent) are multi-sets of formulas in L(C).A (sequent) rule is a pair 〈Θ1, . . . ,Θn, Ω〉, indicated by

Θ1 . . . Θn

Ω,

where Θ1, . . . , Θn (the premises) and Ω (the conclusion) are sequents. A sequentcalculus is a pair 〈C, R〉, where C is a signature and R is a set of rules includingstructural rules and specific rules for the connectives.

• Structural rules:

ξ1, ∆1 → ∆2 ∆1 → ∆2, ξ1

∆1 → ∆2Cut

∆1 → ∆2

∆1 → ∆2, ξ1RW

∆1 → ∆2

ξ1,∆1 → ∆2LW

∆1 → ξ1, ξ1, ∆2

∆1 → ξ1, ∆2RC

∆1, ξ1, ξ1 → ∆2

∆1, ξ1 → ∆2LC

• Left rules: the conclusion has the form c(ϕ1, . . . , ϕn), ∆1 → ∆2 for somen-ary connective c.

• Right rules: the conclusion has the form ∆1 → ∆2, c(ϕ1, . . . , ϕn) for somen-ary connective c.

A sequent s is derivable from a set of sequents H, denoted by H `G s, if thereis a finite sequence (a derivation) of sequents ∆1,1 → ∆2,1 . . .∆1,n → ∆2,n suchthat:

• ∆1,1 → ∆2,1 is s;

• for each i = 1, . . . , n, one of the following holds:

– ∆1,i → ∆2,i is an axiom (justified by Ax), that is ∆1,i ∩∆2,i 6= ∅;

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– ∆1,i → ∆2,i ∈ H (justified by Hyp);

– there exist a rule r = 〈Θ1, . . . ,Θk,Ω〉 and a substitution σ suchthat ∆1,i → ∆2,i = σ(Ω) and σ(Θj) ∈ ∆1,i+1 → ∆2,i+1, . . .∆1,n →∆2,n for j = 1, . . . , k (justified by r and the indexes of σ(Θj)).

We say that a formula ϕ is sequent-derivable from the set of formulas Γ, indi-cated by Γ `G ϕ, if `G Γ → ϕ. The following result is easy to prove.

Proposition 2.5 A sequent calculus G induces a consequence system C(G) =〈C,`G〉 such that Γ`G = ϕ : Γ `G ϕ. Furthermore, C(G) is compact andclosed for substitution.

Example 2.6 A sequent calculus GS4 for modal logic S4 (characterized byreflexive and transitive frames) has the following specific rules, as presented inpage 287 of [25], where (¤∆) is (¤δ) : δ ∈ ∆ and (♦∆) is (♦δ) : δ ∈ ∆:

∆1, ξ1 → ξ2, ∆2

∆1 → (ξ1 ⇒ ξ2), ∆2R⇒ ∆1 → ξ1, ∆2 ∆1, ξ2 → ∆2

∆1, (ξ1 ⇒ ξ2) → ∆2L⇒

∆1, ξ1 → ∆2

∆1 → (¬ ξ1),∆2R¬ ∆1 → ξ1, ∆2

∆1, (¬ ξ1) → ∆2L¬

(¤∆1) → ξ1, (♦∆2)∆′

1, (¤∆1) → (¤ξ1), (♦∆2), ∆′2

R¤ ∆1, ξ1, (¤ξ1) → ∆2

∆1, (¤ξ1) → ∆2L¤

Observe that weakening and contraction can be derived from these rules. Asan example, we can derive → ξ1 `GS4

→ (¤ξ1). Note that we can extracta sequent calculus for propositional logic by eliminating rules R¤ and L¤. Itis also easy to get the right and left rules for ♦ using the abbreviation (♦ϕ) is(¬(¤(¬ϕ))). /

2.2.3 Tableau calculi

Most of the tableau calculi rely on the existence of a negation in the logicat hand. To be able to deal with as many logics as possible we avoid thisassumption by considering tableau calculi over labelled formulas. Herein weonly consider a very simple case for the labels. A labelled formula is a pair〈ϕ, i〉, indicated by i : ϕ, where i is either 0 or 1. Intuitively speaking, 1 : ϕstates that we want ϕ to be true and 0 : ϕ means that we want ϕ to be false.We denote by Lλ the set of pairs i : ϕ such that i = 0, 1 and ϕ ∈ L(C). A(tableau) rule is a pair 〈Υ, µ〉 where Υ ∈ ℘fin℘finL

λ(C) and µ ∈ Lλ(C). Theformula µ is the conclusion of the rule and each set in Υ is said to be analternative. We can look at Υ as alternatives to µ. A tableau calculus is a pair〈C,R〉 where C is a signature and R is a set including the following rules:

• EM (excluded middle): 〈ϕ, 1:ξ, ϕ, 0:ξ, ϕ〉;• for each connective c, a positive rule with conclusion 1:c(ϕ1, . . . , ϕk) and

a negative rule with conclusion 0:c(ϕ1, . . . , ϕk).

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Rule EM is not always included in the definition of a tableau calculus, but itcan easily be shown to be admissible (see [2] for details) if the tableau system isknown to be complete, otherwise the proof of admissibility is not straightforward(coinciding with cut-elimination in sequent calculi). We chose to include it inour definition as it makes the work further on easier.

A tableau-derivation of a set of labelled formulas Θ from a set H of sets oflabelled formulas is a sequence Ψ1 . . .Ψn of finite sets of labelled formulas suchthat:

• Ψ1 is Θ;

• for each i = 1, . . . , n, one of the following holds:

– there is a ψ ∈ L(C) such that 1:ψ, 0:ψ ∈ Ψi (justified by Abs);

– Ψi ∈ H (justified by Hyp);

– there exist a substitution σ and a rule r = 〈Υ, µ〉 such that σ(µ) ∈ Ψi

and for each υ ∈ Υ, σ(υ) ∪ (Ψi \ σ(µ)) ∈ Ψi+1, . . . , Ψn (justifiedby r).

We say that a formula ϕ is tableau-derivable from the finite set of formulas ∆,indicated by ∆ `S ϕ, if `S (1 :δ) : δ ∈ ∆ ∪ 0:ϕ. We say that ϕ is derivablefrom Γ (not necessarily finite), Γ `S ϕ, if ∆ `S ϕ for some finite ∆ ⊆ Γ. Thefollowing result is easy to prove.

Proposition 2.7 A tableau calculus S induces a consequence system C(S) =〈C,`S〉 such that Γ`S = ϕ : Γ `S ϕ. Furthermore, C(S) is compact andclosed for substitution.

Example 2.8 A tableau calculus SP∧,⇒ for the propositional connectives ∧and ⇒ has the following specific rules:

1:ξ1, 1:ξ21:(ξ1 ∧ ξ2)

1 ∧ 0:ξ1 0:ξ20:(ξ1 ∧ ξ2)

0∧

0:ξ1 1:ξ21:(ξ1 ⇒ ξ2)

1⇒ 1:ξ1, 0:ξ20:(ξ1 ⇒ ξ2)

0⇒

Observe that the rule 0∧ states that there are two alternatives for a conjunctionto be false. /

2.2.4 Interpretation structures

Now we show that semantic structures also induce consequence systems. Aninterpretation structure is a triple S = 〈C,M,°〉 where C is a signature, M isa class and °⊆ M × L(C). The elements of the class are called models and °is the satisfaction relation. We denote by Mod(ϕ) the set m ∈ M : m ° ϕand by Mod(Γ) =

⋂

γ∈Γ

Mod(γ).

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Proposition 2.9 The interpretation structure S induces a system C(S) =〈C, ²〉 where Γ² = ϕ ∈ L(C) : Mod(Γ) ⊆ Mod(ϕ) that has all the prop-erties of a consequence system with the possible exception of the closure underrenaming substitutions.

An interpretation system S is sensible-to-renaming if for each renamingsubstitution ρ there is a map βρ : M → M such that m ° ρ(ϕ) iff βρ(m) ° ϕ.

Proposition 2.10 Let S be a sensible-to-renaming interpretation structure.Then C(S) = 〈C, ²〉 is a consequence system.

Proof: Let ρ be a renaming substitution and ϕ ∈ Γ². Then Mod(Γ) ⊆ Mod(ϕ).Assume that m ∈ Mod(ρ(Γ)). Then βρ(m) ∈ Mod(Γ), hence βρ(m) ∈ Mod(ϕ)and so m ∈ Mod(ρ(ϕ)). ¦

Example 2.11 The (Kripke) interpretation system SB for modal logic withaxiom B (sound with respect to symmetric frames) is as follows:

• C is the same as in Example 2.4;

• each model is a tuple (Kripke structure) 〈W,R, V 〉 where W is a non-empty set, R ⊆ W ×W is a binary relation symmetric and transitive andV : Ξ → ℘W is a map;

• m ° ϕ if m, w ° ϕ for every w ∈ W , where:

– m,w ° ξ if w ∈ V (ξ);

– m,w ° (¬ϕ) if not m,w ° ϕ;

– m,w ° (ϕ1 ∧ ϕ2) if m,w ° ϕ1 and m,w ° ϕ2;

– m,w ° (¤ϕ) if m,u ° ϕ for every u ∈ W such that wRu.

The induced C(SB) is a consequence system. Indeed SB is sensible-to-renaming.Let ρ be a renaming substitution and let βρ(〈W,R, V 〉) = 〈W,R, V ′〉 whereV ′ = V ρ. It is easy to prove by induction on the structure of the formula ϕthat 〈W,R, V 〉, w ° ρ(ϕ) iff 〈W,R, V ′〉, w ° ϕ. /

2.3 Fibring

The language of the fibring of two consequence systems will be generated bythe union of the connectives in both signatures. An essential ingredient for thedefinition of the consequence relation will be the ability to translate formulasof the fibring to either component. We achieve this by renaming the schemavariables and coding formulas by fresh variables.

Assume that C ⊆ C ′ and let g : L(C ′) → N be a bijection. The translation

τg : L(C ′) → L(C)

is a map defined inductively as follows:

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• τg(ξi) = ξ2i+1 for ξi ∈ Ξ;

• τg(c) = c for c ∈ C0;

• τg(c(γ′1, . . . , γ′k)) = c(τg(γ′1), . . . , τg(γ′k)) for c ∈ Ck and γ′1, . . . , γ

′k ∈ L(C ′);

• τg(c′(γ′1, . . . , γ′k)) = ξ2g(c′(γ′1,...,γ′k)) for c′ ∈ C ′

k \Ck and γ′1, . . . , γ′k ∈ L(C ′).

Observe that looking at the index of a variable in τ(L(C ′)) we can decidewhether it comes from a variable or a formula starting with a connective inC ′ \ C.

On the other hand, let τ−1g : Ξ → L(C ′) be the following substitution:

• τ−1g (ξ2i+1) = ξi for ξi ∈ Ξ;

• τ−1g (ξ2i) = g−1(i).

In the sequel, we fix a bijective map g and omit the reference to g and justwrite τ . The following lemma is proved using a straightforward induction.

Lemma 2.12 If C ⊆ C ′, then τ−1 τ = id and τ τ−1 = id.

We are ready to define fibring of consequence systems. We start with somenotation. Assume that C ⊆ C ′ and 〈C,`〉 is a consequence system. For eachΓ′ ⊆ L(C ′) we define its closure as follows: Γ′` = τ−1(τ(Γ′)`), where τ isthe translation map from L(C ′) to L(C). We assume given a bijective mapg : L(C ′) ∪ L(C ′′) → N and denote by τ ′g : L(C ′) ∪ L(C ′′) → L(C ′) andτ ′′g : L(C ′) ∪ L(C ′′) → L(C ′′) the corresponding translation maps. We use τ ′

and τ ′′ instead of τ ′g and τ ′′g .The fibring of consequence systems C′ = 〈C ′,`′〉 and C′′ = 〈C ′′,`′′〉 is a pair

C′ ] C′′ = 〈C,`〉 where

• C = C ′ ∪ C ′′;

• `: ℘L(C) → ℘L(C) where, for each Γ ⊆ L(C), Γ` is inductively definedas follows:

1. Γ ⊆ Γ`;

2. If ∆ ⊆ Γ` then ∆`′ ∪∆`′′ ⊆ Γ`;

using the translation maps τ ′ and τ ′′.

The fibring is said to be unconstrained when C ′ ∩ C ′′ = ∅; otherwise is con-strained. Fibring can be seen as a “limit” construction over the class of quasiconsequence systems.

Proposition 2.13 Consider the following transfinite sequence of quasi conse-quence systems:

• C0 = 〈C ′ ∪ C ′′,`0〉 where Γ`0 = Γ for every Γ ⊆ L(C);

• Cβ+1 = 〈C ′ ∪C ′′,`β+1〉 where Γ`β+1 = τ ′−1(τ ′(Γ`β )`′) ∪ τ ′′−1(τ ′′(Γ`β )`′′)for every Γ ⊆ L(C);

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• Cα = 〈C ′ ∪ C ′′,`α〉 where Γ`α =⋃

β<α Γ`β if α is a limit ordinal.

Then C′ ] C′′ = Cα for some ordinal α.

Proof: The operator Υ : ℘L(C) → ℘L(C) such that Υ(∆) = ∆`′ ∪ ∆`′′

is monotonic and extensive, that is, Γ ⊆ Υ(Γ), over the complete lattice〈℘L(C),⊆〉. Hence Υ satisfies Tarski’s fixed point theorem and so, for eachΓ, there is a least fixed point Γ`α . It is easy to see that Γ`α = Γ`. ¦

A sufficient condition can be given stating when this construction is finite.

Proposition 2.14 Let C′ and C′′ be compact consequence systems. Then

C′ ] C′′ =⋃

i∈NCi.

Proof: It is enough to show that the operator Υ : ℘L(C) → ℘L(C) defined inProposition 2.13 is continuous w.r.t. the same order as before, and so Kleene’sfixed point theorem can be applied. The operator Υ is continuous if it preservesdirected unions. Let ∆aa∈A be a family of sets in L(C) with A a directed set.Monotonicity implies that

⋃

a∈A

Υ(∆a) ⊆ Υ(⋃

a∈A

∆a).

It remains to show the other inclusion. Let ϕ ∈ Υ(⋃

a∈A ∆a). Assume, withno loss of generality, that ϕ ∈ (

⋃a∈A ∆a)`

′. Then ϕ ∈ τ ′−1(τ ′(

⋃a∈A ∆a)`

′)

and so there is ϕ′ ∈ (τ ′(⋃

a∈A ∆a))`′

such that τ ′−1(ϕ′) is ϕ. Since 〈C ′ `′〉is compact there is B ⊆ A finite such that ϕ′ ∈ (τ ′(

⋃b∈B ∆b))`

′. Since A

is a directed set, there is d ∈ A such that ∆b ⊆ ∆d for every b ∈ B, andso ϕ′ ∈ τ ′(∆d). Therefore ϕ ∈ τ ′−1(τ ′(∆d)). Applying Kleene’s fixed pointtheorem, we conclude that Γ` =

⋃n∈NΥn(Γ). ¦

In general, we can still place an upper bound on the cardinality of α.

Proposition 2.15 With the notation of Proposition 2.13, α is countable.

Proof: The sequence Γ`0 , Γ`1 , . . . ,Γ`α is strictly increasing, hence |Γ`β | ≥ |β|for each β = 0, . . . , α. Since Γ` ⊆ L(C) and L(C) is countable, it follows thatα must also be countable. ¦

We now show that the fibring of two consequence systems is a consequencesystem and moreover that the consequence operator is related with the conse-quence operators of the original consequence systems.

Proposition 2.16 Fibring C′ ] C′′ is a consequence system. Moreover

C′ ≤ C′ ] C′′ and C′′ ≤ C′ ] C′′.

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Proof: (a) C′ ] C′′ is a consequence system.

(1) Extensivity: Γ ⊆ Γ`. Let γ ∈ Γ. Then τ ′(γ) ∈ τ ′(Γ). By exten-sivity of `′, τ ′(γ) ∈ (τ ′(Γ))`′ ; thus τ ′−1(τ ′(γ)) ∈ τ ′−1((τ ′(Γ))`′), henceγ ∈ τ ′−1((τ ′(Γ))`′) using Lemma 2.12. Therefore γ ∈ Γ`′ and so γ ∈ Γ`.

(2) Monotonicity. Let Γ1 ⊆ Γ2. Then τ ′(Γ1) ⊆ τ ′(Γ2) and τ ′′(Γ1) ⊆ τ ′′(Γ2),and by monotonicity of `′ and `′′, (τ ′(Γ1))`

′ ⊆ (τ ′(Γ2))`′and (τ ′′(Γ1))`

′′ ⊆(τ ′′(Γ2))`

′′. Hence Γ`1 ⊆ Γ`2 .

(3) Idempotence. By definition of ` there is α such that (Γ`)` = (Γ`)`α . Weshow by induction that (Γ`)`α ⊆ Γ` for every α. (i) α = 0. Obvious.(ii) α = β + 1. By induction hypothesis (Γ`)`β ⊆ Γ` and so, by definitionof Γ`, we have (((Γ`)`β )`′ ∪ ((Γ`)`β )`′′) ⊆ Γ` which leads, by definition of`, to (Γ`)`α ⊆ Γ`. (iii) α is a limit ordinal. Straightforward.

(4) Closure for renaming substitutions. Let ρ be a renaming substitution. Foreach Γ ⊆ L(C ′ ∪ C ′′) there is α such that Γ` = Γ`α . It is enough to proveby induction on α that ρ(Γ`α) ⊆ (ρ(Γ))`. (i) α = 0. Then ρ(Γ) ⊆ (ρ(Γ))`

by extensivity. (ii) α = β + 1. Then ρ(Γ`α) = ρ((Γ`β )`′ ∪ (Γ`β )`′′) ⊆(ρ(Γ`β ))`′ ∪ (ρ(Γ`β ))`′′ since C′ and C′′ are closed for renaming substitu-tions. By induction hypothesis, ρ(Γ`β ) ⊆ (ρ(Γ))` and so both (ρ(Γ`β ))`′ ⊆(ρ(Γ))` and (ρ(Γ`β ))`′′ ⊆ (ρ(Γ))`. (iii) α is a limit ordinal. Straightfor-ward.

(b) Since C ′ ⊆ C ′ ∪ C ′′, it remains to show that Γ′`′ ⊆ Γ′` for Γ′ ⊆ L(C ′),and similarly for C′′. Assume that ϕ′ ∈ Γ′`′ . Let ρ : Ξ → L(C ′) be a renamingsubstitution such that ρ(ξi) = ξ2i+1 for every ξi ∈ Ξ. Since C′ is closed forrenaming substitutions, ρ(ϕ′) ∈ (ρ(Γ′))`′ . Observing that ρ coincides with τ ′

for formulas in L(C ′), we have τ ′(ϕ′) ∈ (τ ′(Γ′))`′ , hence ϕ′ ∈ τ ′−1((τ ′(Γ′))`′)and so ϕ′ ∈ Γ`. ¦

The following result shows that the closure in C′ ] C′′ of a set of formulasΓ′ in L(C ′) is the same as the closure in C′ of Γ′. The same applies to C′′. Aspointed out by Gabbay in [12], this is a key requirement for a good definitionof fibring.

Proposition 2.17 Unconstrained fibring C′ ] C′′ of non-trivial consequencesystems closed under substitution is conservative, that is Γ′` = Γ′`′ for everyΓ′ ⊆ L(C ′) and Γ′′` = Γ′′`′′ for every Γ′′ ⊆ L(C ′′).

Proof: There is α such that Γ′` = Γ′`α . We show by induction that Γ′`α ⊆Γ′`′ for every α. (i) α = 0. Then Γ′ ⊆ Γ′`′ by extensivity of `′. (ii) α =β + 1. We have two cases. (1) ϕ′ ∈ τ ′−1((τ ′(Γ′`β ))`′). Since C′ is closed undersubstitution, it follows that τ ′−1((τ ′(Γ′`β ))`′) ⊆ (τ ′−1((τ ′(Γ′`β ))))`′ , hence ϕ′ ∈(τ ′−1((τ ′(Γ′`β ))))`′ and by Lemma 2.12, ϕ′ ∈ (Γ′`β )`′ . On the other hand, bythe induction hypothesis, Γ′`β ⊆ Γ`′ and so monotonicity and idempotence of`′, ϕ ∈ Γ′`′ . (2) ϕ′ ∈ τ ′′−1((τ ′′(Γ′))`′′). Then there is ϕ′′ ∈ (τ ′′(Γ′))`′′ suchthat τ ′′−1(ϕ′′) is ϕ′. Observe that, since C ′ ∩ C ′′ = ∅, ϕ′′ must be a variable.

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Since C′′ is non-trivial, the only variables in τ ′′−1((τ ′′(Γ′))`′′) are those alreadyin τ ′′(Γ′), so ϕ′′ ∈ τ ′′(Γ′) and hence ϕ′ ∈ Γ′. ¦

Fibring plays a special in the class of consequence systems as stated inbelow. We need before an auxiliary result.

Proposition 2.18 The fibring of consequence systems that are closed for sub-stitution is also closed for substitution.

Proof: The basic step is to show by induction that σ(Γ`α) ⊆ (σ(Γ))`. ¦

Proposition 2.19 Fibring is the supremum in the class of consequence systemsclosed for substitution.

Proof: By Propositions 2.16 and 2.18, one only has to prove that C′]C′′ ≤ C′′′whenever C′ ≤ C′′′ and C′′ ≤ C′′′. Let C′′′ be such a system; we have to showthat Γ` ⊆ Γ`′′′ . Since there is α such that Γ` = Γ`α we show by inductionthat Γ`α ⊆ Γ`′′′ for every α. (i) α = 0. Then Γ ⊆ Γ`′′′ , by extensivity of`′′′. (ii) α = β + 1. We have to show that τ ′−1(τ ′(Γ`β ))`′ ⊆ Γ`′′′ and simi-larly τ ′′−1(τ ′′(Γ`β ))`′′ ⊆ Γ`′′′ . By the induction hypothesis Γ`β ⊆ Γ`′′′ , henceτ ′(Γ`β ) ⊆ τ ′(Γ`′′′) and so, by monotonicity of `′, (τ ′(Γ`β ))`′ ⊆ (τ ′(Γ`′′′))`′ .But C′ ≤ C′′′ by hypothesis hence (τ ′(Γ`′′′))`′ ⊆ (τ ′(Γ`′′′))`′′′ and therefore(τ ′(Γ`β ))`′ ⊆ (τ ′(Γ`′′′))`′′′ . Moreover τ ′−1((τ ′(Γ`β ))`′) ⊆ τ ′−1((τ ′(Γ`′′′))`′′′),then, since C′′′ is closed for substitution, τ ′−1((τ ′(Γ`β ))`′) ⊆ (τ ′−1(τ ′(Γ`′′′)))`′′′ ,hence τ ′−1((τ ′(Γ`β ))`′) ⊆ (Γ`′′′)`′′′ and so τ ′−1(τ ′(Γ`β ))`′ ⊆ Γ`′′′ , by idempo-tence of `′′′. (iii) The case where α is a limit ordinal is straightforward. ¦

Now we can give a first attempt to solve the problem of heterogeneous fibringat the deductive level. The basic idea is that once we are given two calculi, ofthe same kind or not, we extract the induced consequence systems thus gettinga homogeneous scenario. Afterwards we obtain the consequence system thatrepresents the fibring of the induced consequence systems.

Example 2.20 Assume that we start with a Hilbert calculus H ′ and a sequentcalculus G′′. Their fibring is the consequence system

C(H ′) ] C(G′′).

Note that the notion of derivation (finite sequence of either formulas or se-quents) does not play a role in the construction. /

From a deductive point of view this solution to heterogeneous fibring is notentirely acceptable because the central notion of derivation as a finite sequenceis lost. We do not have the notion of derivation in the fibring. Therefore weprefer to introduce, in the next section, the new notion of proof system to solvethe problem of fibring heterogeneous proof system.

However, fibring of consequence systems is a good abstraction if we wantto combine a semantic system with a deductive calculus as in the followingexample.

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Example 2.21 Let GS4 be the sequent calculus for modal logic S4 as presentedin Example 2.6 with C ′

1 = ¬, ¤′, C ′2 = ⇒ and C(GS4) = 〈C ′,`′〉 be the

induced consequence system. Let SB be interpretation system over C ′′, whereC ′′

1 = ¬, ¤′′, C ′′2 = ⇒ as introduced in Example 2.11. Observe that the

propositional connectives are the same but we have two different necessitationoperators. Let C(SB) = 〈C ′′,²〉 be the induced consequence system. The fibringof GS4 and SB is the consequence system

C(GS4) ] C(SB) = 〈C ′ ∪ C ′′,`〉.For instance (¤′(¤′′(¤′′(♦′′(♦′(ξ1 ⇒ (¤′ξ1))))))) ∈ ∅`. Indeed:

• (♦′(ξ1 ⇒ (¤′ξ1))) ∈ ∅`′ and so (♦′(ξ1 ⇒ (¤′ξ1))) ∈ ∅`1 ;

• (¤′′(¤′′(♦′′(ξi)))) ∈ ξi² and so (¤′′(¤′′(♦′′(♦′(ξ1 ⇒ (¤′ξ1)))))) ∈ ∅`2 ;

• (¤′ξj) ∈ ξj`′ and so (¤′(¤′′(¤′′(♦′′(♦′(ξ1 ⇒ (¤′ξ1))))))) ∈ ∅`3 ;

• (¤′(¤′′(¤′′(♦′′(♦′(ξ1 ⇒ (¤′ξ1))))))) ∈ ∅` since ∅`3 ⊆ ∅`;with ξi = τ ′′((♦′(ξ1 ⇒ (¤′ξ1)))) and ξj = τ ′((¤′′(¤′′(♦′′(♦′(ξ1 ⇒ (¤′ξ1))))))).

Of course one may ask if fibring the consequence systems induced by twointerpretation structures is the best solution for solving the problem of hetero-geneous fibring at the semantic level. The answer is that there are better ways,namely following an algebraic semantic approach as in [21].

Finally we discuss some preservation results such as preservation of com-pactness and semi-decidability.

Theorem 2.22 The fibring of compact consequence systems is also a compactconsequence system.

Proof: We have to show that Γ` =⋃

Φ∈℘finΓ Φ`. (1) The inclusion from rightto left follows directly by extensivity and monotonicity. It remains to prove theinclusion from left to right. (2) Let ϕ ∈ Γ`. We prove by induction on α that ifϕ ∈ Γ`α then there is Φ ⊆ Γ finite such that ϕ ∈ Φ`. (i) α = 0. Then ϕ ∈ Γ`0 ,hence ϕ ∈ Γ and so we can take Φ = ϕ. (ii) α = β + 1. Then we have twocases. Without loss of generality, let ϕ ∈ (Γ`β )`′ . Since `′ is compact there isΨ ⊆ Γ`β finite such that ϕ ∈ Ψ`′ and moreover by definition of ` also ϕ ∈ Ψ`.But, by the induction hypothesis, for each ψ ∈ Ψ there is Φψ ⊆ Γ finite suchthat ψ ∈ Φ`ψ. Take Φ =

⋃ψ∈Ψ Φψ. Then Φ is a finite set such that Φ ⊆ Γ and

ϕ ∈ Φ`. (iii) The case where α is a limit ordinal is straightforward. ¦

We cannot prove preservation of semi-decidability in general. However, wecan assume a very generic hypothesis satisfied by any “reasonable” consequencesystem; in other words, counter-examples will be very strange looking systems.

Theorem 2.23 Let 〈C ′,`′〉 and 〈C ′′,`′′〉 be strongly semi-decidable conse-quence systems and let C = 〈C,`〉 be their fibring. Assume that, for eachrecursively enumerable Γ ⊆ L(C), Γ` = Γ`α for some α < ωCT

1 , where ωCT1

is the Church–Kleene ordinal [6] (in other words, α is recursively enumerable).Then C is a strongly semi-decidable consequence system.

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Proof: Let Γ ⊆ L(C ′)∪L(C ′′) be a recursively enumerable set. By hypothesisthere is a recursively enumerable α such that Γ` = Γ`α ; we show that Γ`β

is recursively enumerable for every β ≤ α by induction. (i) α = 0. ThenΓ0 = Γ, which is by hypothesis a recursively enumerable set. (ii) Assume Γ`β

is a recursively enumerable set. Since τ ′ is recursive, τ ′(Γ`β ) is also recursivelyenumerable, and so τ ′(Γ`β )`′ is recursively enumerable, since 〈C ′,`′〉 is stronglysemi-decidable. Finally τ ′−1 is again recursive, so τ ′−1(τ ′(Γ`β )`′) is recursivelyenumerable. The same reasoning shows that τ ′′−1(τ ′′(Γ`β )`′′) is recursivelyenumerable, and since the union of two recursively enumerable sets is recursivelyenumerable we conclude that Γ`β+1 is recursively enumerable. Finally, let ε ≤ αbe a limit ordinal. Then Γ`ε =

⋃β<ε Γ`β . Since ε is recursively enumerable,

there is a recursive enumeration of β : β < ε, and hence Γ`ε is a recursiveunion of recursively enumerable sets and therefore is a recursively enumerableset. ¦

Corollary 2.24 The fibring of semi-decidable compact consequence systems isa semi-decidable consequence system.

Proof: By Proposition 2.2, if C′ and C′′ are semi-decidable and compact, thenthey are both strongly semi-decidable. By Theorem 2.23 (which we can applysince in this case α = ω by Proposition 2.14), their fibring C = C′ ] C′′ is alsostrongly semi-decidable, hence in particular semi-decidable. ¦

3 Abstract proof systems

The section is dedicated to (abstract) proof systems, which abstract from theusual syntactic presentations of logics keeping the notion of derivation or cer-tificate.

The notion we present is intentionally very abstract. A proof system (de-fined below) is simply a set of formulas together with a set of derivations aboutwhich very little is assumed. Thus, derivations can be sequences of formulaswith some internal structure (as is the case, for example, in Hilbert calculi) orbear little or no relationship with the language. We show two examples of thelatter, one where derivations are natural numbers and another where there isjust one derivation.

A consequence of having such a general definition is that we cannot statein general properties of the (eventual) structure of the derivations and analyzetheir preservation through fibring. We do not feel this to be a problem at thisstage, however, since we are dealing with a more basic question – namely, how toconstruct a heterogeneous fibring of two logics. Other properties (for example,size of the derivations) can also be studied by looking at subclasses of proofsystems generated by specific mechanisms; we intend to do this in future workon this topic.

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3.1 Basics

Given a binary relation S ⊆ A×B, we will use the notation S(a, b) to indicatethat (a, b) ∈ S or to say that S(a, b) = 1 when viewing the relation as a map1S : A×B → 0, 1.

A proof system is a tuple P = 〈C,D, , P 〉 where C is a signature, D is aset, : ℘(D) ×D → D is a map and P = PΓΓ⊆L(C) is a family of relationsPΓ ⊆ D×L(C) satisfying the following properties, where PΓ(E, Ψ) holds if forevery ψ ∈ Ψ there is e ∈ E such that PΓ(e, ψ) holds:

• Right reflexivity: PΓ(D,Γ) for every Γ ⊆ L(C);

• Monotonicity: PΓ1 ⊆ PΓ2 for every Γ1 ⊆ Γ2 ⊆ L(C);

• Compositionality: Let Γ ∪ ϕ ⊆ L(C):

– ∅ d = d for every d ∈ D;

– If E ⊆ D is a non-empty set and there is Ψ ∈ ℘L(C) such thatPΓ(E,Ψ) and PΨ(d, ϕ) hold then PΓ(E d, ϕ) also holds;

• Variable exchange: PΓ(D, ϕ) = Pρ(Γ)(D, ρ(ϕ)) for any (renaming) substi-tution ρ, that is, a substitution such that ρ(ξ) ∈ Ξ for every ξ ∈ Ξ.

The set D can be seen as the set of possible derivations, is a constructor thatreturns a derivation given a set of derivations and a derivation and, PΓ(d, ψ)holds when d is a derivation of ψ from the set of formulas Γ. A tuple P =〈C,D, , P 〉 is a quasi-proof system if all the properties of a proof system holdwith the possible exception of compositionality.

A particular (though not so interesting) proof system is the one where D =L(C) and P∅(ϕ,ϕ) for every ϕ ∈ L(C). Another example is when we considerD = L(C)∗ (that is, D is the set of all finite sequences of formulas), PΓ(w, γ)if γ ∈ Γ and is the last element of w. We stress that D does not need to berelated to C and to the formulas in L(C); for instance, D can be the set ofnatural numbers. Other examples will be discussed below.

Proposition 3.1 The following properties hold in a proof system:

• Falsehood: PΓ(∅, ϕ) = 0 for every ϕ ∈ L(C);

• Monotonicity on the first argument: PΓ(E1, Ψ) ≤ PΓ(E2, Ψ) for all E1 ⊆E2 ⊆ D and Γ, Ψ ⊆ L(C);

• Anti-monotonicity on the second argument: PΓ(E,Ψ2) ≤ PΓ(E, Ψ1) forall E ⊆ D and Ψ1 ⊆ Ψ2 ⊆ L(C);

• Union: PΓ(E, Ψ1 ∪Ψ2) = PΓ(E, Ψ1)× PΓ(E, Ψ2).

Proof: All the properties follow directly from the definitions. ¦

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Proof systems sometimes have more properties. A proof system is said tobe non-trivial if PΠ(D, ξ) = 0 for any Π ⊆ Ξ and ξ ∈ Ξ \ Π. A proof system isclosed for substitution if PΓ(D,ϕ) ≤ Pσ(Γ)(D, σ(ϕ)) for each substitution σ. Aproof system is said to be compact or finitary if for every Γ and ψ there is Φ ⊆ Γfinite such that PΓ(D,ψ) ≤ PΦ(D, ψ). A proof system is said to be effectiveor decidable if PΓ is recursive for each recursive set Γ ⊆ L(C). It is said to bestrongly effective or strongly decidable if PΓ is recursively enumerable for eachrecursively enumerable set Γ ⊆ L(C).

At this stage, we will not consider the question of whether the set of theo-rems of a given proof system is recursive. Although this is an interesting aspect,our motivation is applications in automated reasoning, where the most impor-tant question is whether a proof can be checked. This is the concept we capturewith the notion of effective proof system.

Proof systems can be related. We say that P = 〈C, D, , P 〉 is weaker thanP ′ = 〈C ′, D′, ′, P ′〉, indicated by P ≤ P ′, if C ⊆ C ′ and PΓ(D, ϕ) ≤ P ′

Γ(D′, ϕ)for every Γ ∪ ϕ ⊆ L(C). Hence a proof system is weaker than another whenit proves less formulas from the same set of formulas.

3.2 Induced proof systems

As in the previous section, we now show that many of the usual syntacticpresentations of deductive systems can be seen to induce proof systems. Thesemantic presentations of logics cannot in general be presented as proof systemsbecause they lack the notion of derivation.

3.2.1 Hilbert calculus

Recall that a Hilbert calculus is a pair 〈C, R〉 such that C is a signature and Ris a set of finitary rules (pairs whose first component is a finite set of formulasand whose second component is a formula). We need some auxiliary notation.Let π(e) denote the last element of sequence e ∈ L(C)∗, by π(E) the set π(e) :e ∈ E where E ⊆ L(C)∗ and by d

π(E)E the sequence obtained by replacing in

d ∈ L(C)∗ the last element of sequence e by sequence e, for every e ∈ E2.As an illustration assume that d is ϕ1 . . . ϕiψkϕi+1 . . . ϕn and E = ψ1 . . . ψk.

Then dπ(E)E is the sequence ϕ1 . . . ϕiψ1 . . . ψkϕi+1 . . . ϕn.

Proposition 3.2 A Hilbert calculus H = 〈C, R〉 induces a compact proof sys-tem P(H) = 〈C,D, , P 〉 as follows:

• D = L(C)∗;

• E d = dπ(E)E ;

• PΓ(d, ψ) holds iff d is a Hilbert-derivation for ψ from Γ.2This is, strictly speaking, not well-defined, since there may be several e with the same

last element. This is not a problem as long as one assumes that e is chosen uniformly, e.g.choosing the first derivation in a lexicographical ordering.

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Proof: (a) Right reflexivity. It follows from the fact that the sequence γ is aderivation of γ from set Γ whenever γ ∈ Γ. (b) Monotonicity. Assume that dis a derivation of ϕ from Γ1 and that Γ1 ⊆ Γ2. Then d is also a derivation ofϕ from Γ2. (c) Compositionality. Assume that PΓ(E, Ψ) and PΨ(d, ϕ) hold forsome set Ψ. Then dΨ

EΨ, where EΨ ⊆ E is the set of derivations in E of each

ψ ∈ Ψ from Γ, is a derivation of ϕ from Γ. Observe that only a finite number ofelements of Ψ are used, so dΨ

EΨis still a finite sequence. (d) Variable exchange.

Assume that d is a derivation of ϕ from Γ and that ρ is a renaming substitution.Then ρ(d) is a derivation of ρ(ϕ) from ρ(Γ). ¦

It is easy to see that P(H) is also closed for substitution translating each deriva-tion with the given substitution.

We give a necessary and sufficient condition for P(H) to be effective.

Theorem 3.3 The proof system P(H) induced by a Hilbert system H = 〈C, R〉such that 〈L(C), g〉 is a Godel domain is effective iff the following relations arerecursive:

• ax ⊆ L(C), such that ax(α) iff α is an instance of an axiom;

• irk+1 ⊆ L(C)k+1 such that irk+1(α1, . . . , αk, β) iff α1, . . . , αk, β is an in-stance of a k + 1-ary inference rule.

Proof: Let hypΓ ⊆ L(C) be such that hypΓ(γ) iff γ ∈ Γ.(1) Assume that PΓ is recursive whenever Γ is a recursive set. In particular, PΦ

is recursive for every finite set Φ.(a) The relation ax is recursive since ax(α) = P∅(〈α〉, α).(b) Let 〈Θ, η〉 be a k-ary rule. The relation irk+1 is recursive since

irk+1(α1, . . . , αk, β) = PΘ(〈α1, . . . , αk, β〉, β)

where Θ = α1, . . . , αk.(2) Let Γ ⊆ L(C) be a recursive set. The relation PΓ such that:

PΓ(ω, β) = (((ω = 〈β〉) ∧ (ax(β) ∨ hypΓ(β)))∨

|ω|∨

k=1

(∃ω′vω(∃i1≤|ω′|(. . . (∃ik≤|ω′|

(PΓ(ω′, last(ω′)) ∧ (ω = ω′ · β) ∧ irk+1((ω′)i1 , . . . , (ω′)ik , β))))))),

where |ω| denotes the length of ω, ω′ v ω stands for “ω′ is a prefix of ω” and· denotes concatenation, is recursive since the relations ax, ir, hyp, length andlast are recursive, being a prefix is decidable, and the conjunction, disjunctionand bounded quantification of recursive relations is a recursive relation. ¦

This is not the only proof system that can be generated from a Hilbertcalculus. By construction any initial segment of a Hilbert derivation is itself avalid derivation, which motivates the following definition.

19

Example 3.4 Let H = 〈C,R〉 be a Hilbert calculus. Then P ′(H) is defined asabove, except that P ′

Γ(d, ϕ) now holds iff d is a valid derivation from Γ and ϕoccurs in d.

The same example can be used to induce a proof system where the set ofderivations bears no (apparent) relationship to the language.

Example 3.5 Let H = 〈C, R〉 be a Hilbert calculus and let g be a Godelizationof L(C), that is, g : N→ L(C) is recursive. Define P ′′(H) as follows:

• D is g(w) : w ∈ L(C)∗;• PΓ(n, ϕ) if n is the Godel number of a sequence w ∈ L(C)∗ and w is a

Hilbert-derivation of ϕ from Γ. /

3.2.2 Sequent calculus

Recall that a sequent calculus is a pair 〈C,R〉 where C is a signature and R isa set of rules (pairs whose first component is a finite set of sequents and whosesecond component is a sequent): structural rules and, for each connective, aright and a left rule. We need some auxiliary notation. Assume that d is asequence of sequents with initial sequent ∆1 → ∆2. When the initial sequentis important we can use d∆1→∆2 to refer to d.

Proposition 3.6 A sequent calculus G = 〈C, R〉 induces a proof system P(G) =〈C,D, , P 〉 defined as follows:

• D = Seq(C)∗ where Seq(C) is the set of all sequents defined with formulasin L(C);

• Let E ∪ dΘ→ϕ ⊆ D where E 6= ∅. Let θ1, . . . , θn ⊆ Θ be the setof all sequents such that dΓi→θi ∈ E for every i = 1, . . . , n. Let Θ =Θ \ θ1, . . . , θn. Then E dΘ→ϕ is the following sequence3:

Θ,Γ1, . . . , Γn → ϕ Cut 1a,1b1a Θ,Γ1, . . . , Γn → ϕ, θ1 LW∗, RW

dΓ1→θ1

1b θ1, Θ, Γ1, . . . ,Γn → ϕ Cut 2a,2b...θ1, . . . , θn−1, Θ, Γ1, . . . ,Γn → ϕ Cut na,nb

na θ1, . . . , θn−1, Θ, Γ1, . . . ,Γn → ϕ, θn LW∗, RWdΓn→θn

nb Θ,Γ1, . . . , Γn → ϕ LW∗

dΘ→ϕ

where LW∗, RW indicate several applications of left weakening followed byright weakening;

3See the footnote on page 18

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• ∅ d = d;

• PΓ(d, ϕ) holds if d is a sequent-derivation of ϕ from Γ.

Proof: (a) Right reflexivity. Just consider the derivation Γ → γ justified asan axiom whenever γ ∈ Γ. (b) Monotonicity. Consider a derivation dΓ1→ϕ andΓ1 ⊆ Γ2. Then the following is a derivation for Γ2 → ϕ:

1 Γ2 → ϕ LW∗ 22 dΓ1→ϕ

(c) Compositionality. Direct from the definition of . (d) Variable exchange.If d is a derivation for Γ → ϕ and ρ is a renaming substitution then ρ(d) is aderivation for ρ(Γ) → ρ(ϕ). ¦

Remark 3.7 Observe that in the case of sequents we can define binary rela-tions PH ⊆ D×Seq(C) where H is a set of sequents over L(C) (hence in Seq(C))but stating that PH(d, s) = 1 whenever H `G s with sequent-derivation d. Ofcourse PΓ(d, ϕ) is P∅(d,Γ → ϕ).

3.2.3 Tableau calculus

Recall that a tableau calculus is a pair 〈C, R〉 where C is a signature and Ris a set of rules (pairs whose first component is a set of finite sets of labelledformulas and whose second element is a labelled formula): excluded middle andfor each connective a positive and a negative rule. We need some notation: ifd1 . . . dn is a finite sequence of sets and Ψ is a set of labelled formulas, thenΨd1 . . . dn is the finite sequence d1 ∪ Ψ . . . dn ∪ Ψ. Observe that if d1 . . . dn

is a tableau-derivation for Γ `S ϕ then Ψd1 . . . dn is a tableau-derivation forΓ ∪Ψ `S ϕ. We use 1:A to refer to the set (1 :a) : a ∈ A.Proposition 3.8 A tableau calculus G = 〈C,R〉 induces a proof system P(G) =〈C,D, , P 〉 defined as follows:

• D = (℘Lλ(C))∗ is the set of all finite sequences of sets of labelled formulas;

• Let E ∪ d ⊆ D be such that the first set in d is 1 :Θ ∪ 0 :ϕ and, forθi ∈ Θ, with i = 1, . . . , n, there are ei ∈ E whose first set is 1 :Γi∪0:θi.Let Θ = Θ \ θ1, . . . , θn and Γ =

⋃ni=1 Γi. Define E d as follows, where

Ψ1 is 1 : Θ ∪ (1 : Γ \ 1 : Γ1) ∪ 0 : ϕ and Ψn is 1 : Θ ∪ (1 : Γ \ 1 : Γn) ∪ 1 :θ1, . . . , 1:θn−1, 0:ϕ:

1 : Θ, 1:Γ, 0:ϕ EM 1a,1b1a 0:θ1, 1:Θ, 1:Γ, 0:ϕ

Ψ1e1

1b 1:θ1, 1:Γ, 0:ϕ EM 2a,2b...1 :θ1, . . . , 1:θn−1, 1:Γ, 0:ϕ EM na,nb

na 1:θ1, . . . , 0:θn, 1:Γ, 0:ϕΨn

nb 1:Θ, 0:ϕ1:Γd

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• PΓ(d, ϕ) holds iff d is a tableau derivation for ϕ from Γ.

Proof: (a) Right reflexivity. The set 1 : Γ ∪ 0 : γ is an absurd when γ ∈ Γ.(b) Monotonicity. Let Γ1 ⊆ Γ2 and d be a tableau-derivation of ϕ from Γ1. Then(Γ2 \ Γ1)d is a tableau-derivation of ϕ from Γ2. (c) Compositionality. Followsfrom the fact that E d is a tableau-derivation of ϕ from Θ∩Γ whenever d is atableau-derivation from Θ and there is a tableau-derivation in E of θi from Γi

for every i = 1, . . . , n. (d) Variable exchange. If d is a tableau-derivation of ϕfrom Γ then ρ(d) is a tableau-derivation of ρ(ϕ) from ρ(Γ) for every renamingsubstitution ρ. ¦

Remark 3.9 Observe that in the case of tableau calculi we can define binaryrelations PH ⊆ D × ℘Lλ(C) where H is a set of sets of labelled formulas overL(C) (hence in ℘Lλ(C)) but stating that PH(d, s) = 1 whenever H `S s withtableau-derivation d. Of course PΓ(d, ϕ) is P∅(d, 1:Γ ∪ 0:ϕ).

3.3 Fibring

The fibring of two proof systems P ′ = 〈C ′, D′, ′, P ′〉 and P ′′ = 〈C ′′, D′′, ′′, P ′′〉is the tuple P ′ ] P ′′ = 〈C,D, , P 〉 defined as follows:

• C = C ′ ∪ C ′′;

• D is a set inductively defined as follows:

– D′ ∪D′′ ⊆ D;

– If E ⊆ D then ℘E × (D′ ∪D′′) ⊆ D;

• E d = 〈E, d〉 if E 6= ∅ and d otherwise;

• PΓ(d′, ϕ) holds if P ′τ ′(Γ)(d

′, τ ′(ϕ)) holds for d′ ∈ D′ ;

• PΓ(d′′, ϕ) holds if P ′′τ ′′(Γ)(d

′′, τ ′′(ϕ)) holds for d′′ ∈ D′′;

• PΓ(〈E, d〉, ϕ) holds iff there is a set Ψ ∈ L(C) for which both PΨ(d, ϕ)and PΓ(E, Ψ) hold.

Notice that the second- and third-to-last cases are not mutually exclusive sinceD′ and D′′ need not be disjoint.

Before showing that the fibring of proof systems is a proof system we give theintuition behind the construction of the set of derivations and some examples.Take as an example 〈d′, d′′〉; this is a derivation provided that d′ is a derivationin D′ and d′′ is a derivation in D′′. Such a derivation is only relevant when weuse the relation P . Saying that

P∅(〈d′, d′′〉, c′′(c′(ξ1)))

holds means:

22

• that d′′ is a derivation of c′′(ξk) where ξk = τ ′′(c′(ξ1)), assuming that wetake the singleton ξk as the set of hypotheses, in other words providedthat P ′′

ξk(d′′, c′′(ξk)) holds;

• and that d′ is a derivation of c′(ξ1) taking the empty set as the set ofhypotheses, in other words provided that P ′

∅(d′, c′(ξ1)) holds.

We now give some examples of fibring involving logics presented with dif-ferent calculi.

Example 3.10 Consider the proof system P(GS4) = 〈C ′, R′〉 induced by thesequent calculus for modal logic S4 as presented in Example 2.6 and the proofsystem P(HB) = 〈C ′′, R′′〉 induced by the Hilbert calculus for modal logic withaxiom B as in Example 2.4. They share the propositional connectives, butin the fibring we have two necessitations: an S4 ¤′ (and consequently an S4diamond ♦′) and a B ¤′′ (and consequently a B diamond ♦′′). We can provein P(GS4) ] P(HB) that

P∅(〈〈d′1, d′′〉, d′2〉, (¤′(¤′′(♦′′(¤′′(♦′(ξ1 ⇒ (¤′ξ1))))))))

holds. Indeed

• P ′ξi(d

′2, (¤′ξi)) holds in P(GS4) with derivation d′2 as follows:

1 → (¤′ξi) R¤′ 22 → ξi Hyp

where ξi is τ ′(¤′′(♦′′(¤′′(♦′(ξ1 ⇒ (¤′ξ1))))));

• and we have to show that P∅(〈d′1, d′′〉, ¤′′(♦′′(¤′′(♦′(ξ1 ⇒ (¤′ξ1))))))holds;

ButP∅(〈d′1, d′′〉, (¤′′(♦′′(¤′′(♦′(ξ1 ⇒ (¤′ξ1)))))))

holds since

• P ′′ξj(d

′′, (¤′′(♦′′(¤′′(ξj))))) holds in P(HB) with derivation d′′ as follows:

1 ξj Hyp2 (¤′′ξj) Nec 13 ((¤′′ξj)⇒ (¤′′(♦′′(¤′′ξj)))) B4 (¤′′(♦′′(¤′′ξj))) MP 2,3

where ξj is τ ′′(♦′(ξ1 ⇒ (¤′ξ1)));

• and P∅(d′1,♦′(ξ1⇒ (¤′ξ1))) holds in P(GS4) with derivation d′1 as follows:

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1 → (♦′(ξ1 ⇒ (¤′ξ1))) R♦′ 22 → (ξ1 ⇒ (¤′ξ1)), (♦′(ξ1 ⇒ (¤′ξ1))) R⇒ 33 ξ1 → (¤′ξ1), (♦′(ξ1 ⇒ (¤′ξ1))) R¤′ 44 → ξ1, (♦′(ξ1 ⇒ (¤′ξ1))) R♦′ 55 → ξ1, (ξ1 ⇒ (¤′ξ1)), (♦′(ξ1 ⇒ (¤′ξ1))) R⇒ 66 ξ1 → ξ1, (¤′ξ1), (♦′(ξ1 ⇒ (¤′ξ1))) Ax

Hence d′2, d′′, d′1 provide a derivation for (¤′(¤′′(♦′′(¤′′(♦′(ξ1 ⇒ (¤′ξ1)))))))

without any hypotheses. Observe that the number of pairings in the derivationindicates the way we have to use the component proof systems. In the exampleabove we have three pairings and we had to use the component proof systemsthree times. /

Example 3.11 Consider the propositional part of the proof system P(GS4) =〈C ′, R′〉 in Example 2.6 (with negation and implication as connectives) andthe proof system C(SP∧,⇒) = 〈C ′′, R′′〉 presented in Example 2.8. We prove inP(GS4) ] P(SP∧,⇒) that

P∅(〈d′1, d′2, d′′〉, ((¬(ξ1 ⇒ ξ2))⇒ (ξ1 ∧ (¬ ξ2))))

holds. Indeed, taking ξi = τ ′′(¬(ξ1 ⇒ ξ2)) and ξj = τ ′′(¬ ξ2), we have:

• P ′′(ξi⇒ξ1),(ξi⇒ξj)(d

′′, (ξi⇒ (ξ1 ∧ ξj))) holds in SP∧,⇒ with derivation d′′ asfollows:

1. 1 :(ξi ⇒ ξ1), 1:(ξi ⇒ ξj), 0:(ξi ⇒ (ξ1 ∧ ξj)) 0⇒ 22. 1 :(ξi ⇒ ξ1), 1:(ξi ⇒ ξj), 1:ξi, 0:(ξ1 ∧ ξj) 1⇒ 3,43. 0 :ξi, 1:(ξi ⇒ ξj), 1:ξi, 0:(ξ1 ∧ ξj) Ax4. 1 :ξ1, 1:(ξi ⇒ ξj), 1:ξi, 0:(ξ1 ∧ ξj) 1⇒ 5,65. 1 :ξ1, 0:ξi, 1:ξi, 0:(ξ1 ∧ ξj) Ax6. 1 :ξ1, 1:ξj , 1:ξi, 0:(ξ1 ∧ ξj) 0∧ 7,87. 1 :ξ1, 1:ξj , 1:ξi, 0:ξ1 Ax8. 1:ξ1, 1:ξj , 1:ξi, 0:ξj Ax

• P∅(d′1, ((¬(ξ1 ⇒ ξ2))⇒ ξ1)) holds in GS4 with derivation d′1 as follows:

1. → ((¬(ξ1 ⇒ ξ2))⇒ ξ1) R⇒ 22. (¬(ξ1 ⇒ ξ2)) → ξ1 L¬ 33. → (ξ1 ⇒ ξ2), ξ1 R⇒ 44. ξ1 → ξ2, ξ1 Ax

• P∅(d′2, (¬(ξ1 ⇒ ξ2))⇒ (¬ ξ2)) holds in GS4 with derivation d′2 as follows:

1. → (((¬(ξ1 ⇒ ξ2))⇒ (¬ ξ2)) R⇒ 22. (¬(ξ1 ⇒ ξ2)) → (¬ ξ2) L¬ 33. → (ξ1 ⇒ ξ2), (¬ ξ2) R⇒ 44. ξ1 → ξ2, (¬ ξ2) R¬ 55. ξ1, ξ2 → ξ2 Ax

24

Hence d′′, d′1, d′2 constitutes a derivation of (¬(ξ1⇒ ξ2))⇒ (ξ1 ∧ (¬ ξ2))) with no

hypotheses. We have two pairings but in one of them we have to produce twoderivations because the corresponding set has two elements. /

The following result shows that the set D in the fibring comes out, as ex-pected, as a fixed point construction.

Proposition 3.12 Consider the transfinite sequence of quasi proof systemswhere D is replaced by Dα.

• D0 = D′ ∪D′′;

• Dβ+1 = Dβ ∪ (℘(Dβ)× (D′ ∪D′′));

• Dα =⋃

β≤α

Dβ where α is a limit ordinal.

Then D = Dα for some ordinal α.

Proof: The operator Υ : L(C) → L(C) such that Υ(∆) = ∆∪(℘∆×(D′∪D′′))is monotonic over the complete lattice 〈℘L(C),⊆〉 and (D′∪D′′) ⊆ Υ(D′∪D′′).Hence Υ satisfies Tarski’s fixed point theorem and so it has a least fixed point.It is easy to see that this fixed point is D. ¦

The question that arises at this point is when can we obtain each element ofD with a finite number of iterations. We give a sufficient condition for that. Forthis purpose we need to work with a special kind of fibring. The finite derivationfibring of proof systems P ′ and P ′′ is the fibring proof system P ′ ]P ′′ where Dis a set inductively defined as follows:

• D′ ∪D′′ ⊆ D;

• If E ⊆ D then (℘finE)× (D′ ∪D′′) ⊆ D;

Proposition 3.13 Let P ′ ] P ′′ be the finite derivation fibring of P ′ and P ′′.Then

D =⋃

i∈NDi.

Proof: The operator Υ′ : L(C) → L(C) such that Υ′(∆) = ∆∪ (℘fin∆× (D′ ∪D′′)), analogous to Υ defined in Proposition 3.12, is continuous and so Kleene’sfixed point theorem can be applied. ¦

We now provide an example of fibring of two proof systems where we needto compute a transfinite fixed point for obtaining the set of derivations andthat is supported by our definition of proof system. This example is admittedlyartificial, but it shows that, whenever infinitary rules are allowed, Dω may notsuffice.

Example 3.14 Consider proof systems P ′ and P ′′ defined as follows:

25

• P ′ is induced by the Hilbert system 〈C,R′〉, where C0 = ϕ, C1 = X, Gand R′ contains the rules rk = 〈X2kϕ, X2k+1ϕ〉;

• P ′′ is the proof system with the same signature C and whose derivationsare either Hilbert derivations from the single rule rk = 〈X2k−1ϕ, X2kϕ〉or a derivation ∗ proving Gϕ from X2n−1ϕ : n ∈ N. We define E d =(E \ ∗) d whenever d 6= ∗ and E ∗ = ∗. Note that this is well definedsince there are no (non-trivial) derivations of the hypotheses needed forto apply ∗.

Intuitively, we can see ϕ as some property of a system we want to model; inP ′ we are given that that property holds at time moment 2k + 1 if it holds astime moment 2k; in P ′′ we state that it holds at moment 2k given that it holdsat moment 2k − 1. The proposition Gϕ states that ϕ holds at all odd timeinstants.

We now show that, in the fibring P = P ′ ] P ′′, Pϕ(d,Gϕ) holds, wherethe derivation d is built in two steps. First we define derivations s′2k and s′′2k+1

by

s′2k: 1 X2kϕ Hyp s′′2k+1: 1 X2k+1ϕ Hyp2 X2k+1ϕ rk 1 2 X2k+2ϕ rk+1 1

noticing that P ′x2kϕ(s

′2k, X

2k+1ϕ) and P ′′x2k+1ϕ(s

′′2k+1, X

2k+2ϕ) hold.We now define

d0 = ϕ (as a sequence)d2k+1 = 〈d2k, s′2k〉d2k+2 = 〈d2n+1, s′′2k+1〉

d = 〈d2k−1 : k ∈ N, ∗〉

and show by induction that Pϕ(dn, Xnϕ) holds. Indeed:

• P ′ϕ(ϕ,ϕ) holds.

• Assuming that Pϕ(dn, Xnϕ) holds, there are two cases. Suppose that n+1 = 2k+1; then the thesis holds because d = 〈d2k, s′2k〉, Pϕ(d2k, X

2kϕ)holds by induction hypothesis and P ′

X2kϕ(s′2k, ϕX2k+1) holds as remarked

above. Suppose that n + 1 = 2k + 2; then the thesis holds becaused = 〈d2k+1, s′′2k+1〉, Pϕ(d2k+1, X

2k+1ϕ) holds by induction hypothesisand P ′′

X2k+1ϕ(s′′2k+1, X

2k+2ϕ) holds.

Finally, from Pϕ(d2k−1 : k ∈ N, X2k−1ϕ : k ∈ N) and P ′′X2k−1ϕ:k∈N(∗, Gϕ)

we conclude that Pϕ(d, Gϕ). It is easy to see that no finite number of stepswill suffice to derive Gϕ from ϕ. /

We investigate the relationship between the fibring and the original proofsystems showing that the latter are weaker than the former. Also of interestis to analyze how the fibring relates with proof systems that are stronger thanthe components.

26

Proposition 3.15 Fibring P ′ ]P ′′ of two proof systems is a proof system andmoreover

P ′ ≤ P ′ ] P ′′ and P ′′ ≤ P ′ ] P ′′.

Proof: (1) We start by proving that P ′ ] P ′′ is a proof system.(i) Right reflexivity. If ϕ ∈ Γ, then τ ′(ϕ) ∈ τ ′(Γ), hence P ′

τ ′(Γ)(d′, τ ′(ϕ)) holds

for some d′ ∈ D′. Therefore PΓ(d′, ϕ) also holds, so PΓ(D, ϕ) holds.(ii) Monotonicity. Suppose Γ1 ⊆ Γ2 and suppose that PΓ1(D, ϕ) holds. Thenthere is α such that PΓ1(Dα, ϕ) holds. We show by induction on α thatPΓ2(Dα, ϕ) holds. (a) α = 0. Without loss of generality, assume that thereis d′ ∈ D′ such that P ′

τ ′(Γ1)(d′, τ ′(ϕ)) holds and, by monotonicity of P ′, also

P ′τ ′(Γ2)(d

′, τ ′(ϕ)) holds, thus PΓ2(d′, ϕ) holds and so PΓ2(D0, ϕ). (b) α =

β + 1. Assume PΓ1(Dβ+1, ϕ) holds. Hence there is Ψ such that PΓ1(Dβ,Ψ)and PΨ(D, ϕ). Using the induction hypothesis we have PΓ2(Dβ, Ψ) and by def-inition of D we get PΓ2(Dα, ϕ). (c) α is a limit ordinal. This case is simple.(iii) Compositionality. Immediate by definition of .(iv) Variable exchange. Let ρ be a renaming substitution and suppose thatPΓ(D, ϕ) holds. Then there is α such that PΓ(Dα, ϕ) holds. We prove by in-duction on α that Pρ(Γ)(Dα, ρ(ϕ)) holds. (a) α = 0. Suppose that d is d′ ∈ D′;then P ′

τ ′(Γ)(d′, τ ′(ϕ)) holds. We have to show that there is e′ ∈ D′ such that

P ′τ ′(ρ(Γ))(e

′, τ ′(ρ(ϕ))) holds. Let ρ′ : Ξ → L(C ′) be the renaming substitu-

tion such that ρ′(ξ) = τ ′(ρ(τ ′−1(ξ))). The variable exchange property for P ′leads to the existence of e′ ∈ D such that P ′

ρ′(τ ′(Γ))(e′, ρ′(τ ′(ϕ))) holds. Since

ρ′(τ ′(ψ)) = τ ′(ρ(ψ)) for every ψ ∈ L(C) we conclude that P ′τ ′(ρ(Γ))(e

′, τ ′(ρ(ϕ)))holds. If d is d′′ ∈ D′′ the proof is similar. (b) α = β + 1. Since PΓ(Dβ+1, ϕ)then there is Ψ such that PΓ(Dβ,Ψ) and PΨ(D,ϕ). For each ψ ∈ Ψ thereexists e ∈ Dβ for which PΓ(e, ψ) holds, and by induction hypothesis, there issome e′(ψ) ∈ Dβ for which Pρ(Γ)(e′(ψ), ρ(ψ)) holds. Thus Pρ(Γ)(Dβ, ρ(Ψ)) forE′ = e′(ψ) : ψ ∈ Ψ. Using a reasoning similar to the one for the basis weconclude that Pρ(Ψ)(D, ρ(ϕ)), and so Pρ(Γ)(Dα, ρ(ϕ)) holds. (c) α is a limitordinal. Straightforward.(2) It remains to show that P ′ ≤ P ′ ] P ′′ and P ′′ ≤ P ′ ] P ′′. Since bothsituations are similar, we show the first one, which amounts to showing thatP ′

Γ′(D′, ϕ′) ≤ PΓ′(D, ϕ′) for Γ′ ∪ ϕ′ ⊆ L(C ′). Suppose that P ′

Γ′(d′, ϕ′) holds.

Then, since τ ′ is a renaming substitution on L(C ′), there is a derivation d ∈ D′

such that P ′τ ′(Γ)(d, τ ′(ϕ)) holds, and therefore PΓ(d, ϕ) holds. ¦

We need an auxiliary result before characterizing fibring in the class of proofsystems that are stronger than the components.

Lemma 3.16 Fibring of proof systems closed for substitution is also closed forsubstitution.

The proof of this result is similar to the one showing that the fibring satisfiesvariable exchange, and for this reason we omit it.

27

Proposition 3.17 Fibring is the supremum in the class of proof systems closedfor substitution.

Proof: By Propositions 3.15 and 3.16, one only has to prove that P ′]P ′′ ≤ P ′′′whenever P ′ ≤ P ′′′ and P ′′ ≤ P ′′′. Let P ′′′ be such a system; we have to showthat PΓ(D,ϕ) ≤ P ′′′

Γ (D′′′, ϕ). Assume that PΓ(D, ϕ) holds. Then there isd ∈ D such that PΓ(d, ϕ). We prove by induction on d that there is d′′′ suchthat P ′′′

Γ (d′′′, ϕ).(i) Assume that d is d′ ∈ D′. Then Pτ ′(Γ)(d′, τ ′(ϕ)) holds and so by the hypoth-esis on P ′′′ there is e′′′ ∈ D′′′ such that P ′′′

τ ′(Γ)(e′′′, τ ′(ϕ)) also holds. Since P ′′′ is

closed for substitution there is d′′′ ∈ D′′′ such that P ′′′τ ′−1(τ ′(Γ))(d

′′′, τ−1(τ ′(ϕ)))holds and so there is d′′′ ∈ D′′′ such that P ′′′

Γ (d′′′, ϕ) holds. If d is d′′ ∈ D′′, thesituation is analogous.(ii) Assume that d is 〈E, f〉. Then there is Ψ ⊆ L(C) such that PΨ(f, ϕ) andPΓ(E, Ψ) hold. By induction hypothesis there is E′′′ such that P ′′′

Γ (E′′′,Ψ)holds. Using a reasoning similar to the one above, there is d′′′ ∈ D′′′ such thatP ′′′

Ψ (d′′′, ϕ) holds. Hence P ′′′Γ (E′′′ d′′′, ϕ) holds. ¦

Now we turn our attention toward preservation of properties by fibringstarting with compactness.

Theorem 3.18 The fibring of compact proof systems is compact.

Proof: We prove that there is Φ ⊆ Γ finite such that PΦ(d, ϕ) wheneverPΓ(d, ϕ) by induction on d.(i) Let d ∈ D′. Then P ′

τ ′(Γ)(d, τ ′(ϕ)), so there are Φ′ ⊆ τ ′(Γ) and d′ ∈ D′ suchthat P ′

Φ′(d′, τ ′(ϕ)) and so Pτ ′−1(Φ′)(d′, ϕ) where τ ′−1(Φ′) ⊆ Γ is finite. The case

d ∈ D′′ is analogous.(ii) Let d = 〈E, d′〉. Then there is Ψ such that PΓ(E, Ψ) and PΨ(d′, ϕ) hold.Using a reasoning similar to the one in the basis, we can conclude that thereare Φ ⊆ Ψ finite and f ∈ D such that PΦ(f, ϕ) holds. On the other hand, sincePΓ(E, Φ) holds, then by induction hypothesis there are Ω ⊆ Γ finite and F ⊆ Dsuch that PΩ(F, Φ), and so PΩ(〈F, f〉, ϕ). ¦

As a consequence, the finite-derivation fibring of compact proof systems has thesame deductive power as their fibring, i.e. the value of PΓ(D, ϕ) is independenton whether D is obtained by fibring or by finite-derivation fibring.

3.4 Abduction

Preservation of effectiveness in the fibring can be investigated. We make use inour reasoning of the Church-Turing postulate. Before that, we make a detourbecause we have to assume a further requirement concerning compositionalityof derivations. We need to be able to “calculate” a set Ψ in the definitionof compositionality. Sometimes we can easily find the set Ψ referred to inthe definition of compositionality, e.g. when we are dealing with proof systems

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induced by known calculi. This motivates the following definition of hypothesis-abductible proof system.

Let P = 〈C, D, , P 〉 be a proof system. An abduction function for P is acomputable function Abd : D → ℘fin℘finL(C) such that for any Γ ⊆ L(C), ifPΓ(d, ϕ) holds then there is ∆ ∈ Abd(d) such that P∆(d, ϕ) and ∆ ⊆ Γ.

A proof system P is said to be hypothesis-abductible if there is an abductionfunction for P.

The intuition is as follows: for d ∈ D, Abd(d) contains sets of hypothesesthat make d a valid derivation in proof system P. It is discussible whether thisfunction should also depend on the formula ϕ one wants to derive. In all ex-amples shown the construction will independent of this formula, supporting theclaim that in general that is also the case; however, it would be straightforwardto extend our results to abduction functions depending on a derivation and aformula.

Proposition 3.19 Every hypothesis-abductible proof system is compact.

Proof: Straightforward from the definition. ¦

Hilbert calculi, sequent calculi and tableau systems all induce hypothesis-abductible proof systems.

Example 3.20 Let H be a Hilbert calculus and P(H) the corresponding in-duced proof system. Then AbdH is an abduction function for P(H), where

AbdH(d) = ϕ : ϕ occurs in d justified by Hyp.

Example 3.21 Let G be a sequent calculus and P(G) the corresponding in-duced proof system. Then AbdG is an abduction function for P(G), where

AbdG(dΓ→∆) = Γ.

Example 3.22 Let S be a tableau calculus and P(S) the corresponding in-duced proof system. Then AbdS is an abduction function for P(S), where

AbdS(d) = ∆ if the first set in d is (1 :∆) ∪ (0 :Φ).

Notice that in all these examples there can be spurrious formulas in theonly set in Abd(d); this is because we do not analyze the derivations except ina syntactical way. In other words, the set of hypotheses generated makes d avalid derivation, but nothing forbids that there be another derivation equivalent(in some sense) to d that depends on a smaller set of formulas.

The examples above suggest that Abd(d) might have been defined to returnsimply a finite set of formulas. However, the fibring of hypothesis-abductibleproof systems would then not be necessarily be hypothesis-abductible. This isillustrated by Example 3.24.

Theorem 3.23 The finite-derivation fibring of hypothesis-abductible proof sys-tems is hypothesis-abductible.

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Proof: Let P ′ and P ′′ be proof systems with abduction functions Abd′ andAbd′′ respectively and let P be their finite-derivation fibring.

Define Abd(d) by recursion on d as follows.

• If d ∈ D′ ∩D′′, then Abd(d) = τ ′−1(Abd′(d)) ∪ τ ′′−1(Abd′′(d)).

• If d ∈ D′, then Abd(d) = τ ′−1(Abd′(d)).

• If d ∈ D′′, then Abd(d) = τ ′′−1(Abd′′(d)).

• If d = 〈E, d0〉, then Abd(d) contains all sets generated as follows. Foreach E′ = e1, . . . , en ⊆ E, let Abd(ei) = Ψi,1, . . . , Ψi,ki. Then Ψ1,i1 ∪Ψ2,i2∪. . .∪Ψn,in ∈ Abd(d) for all values of i1, . . . , in such that the previousexpression makes sense.

Notice that this definition is not really recursive, since the apparent recursivecall in the last case always reduces to a call of Abd′ or Abd′′. Also, since E isfinite, the resulting set is also finite (though possibly very large).

We now show that Abd thus defined is an abduction function. Suppose thatPΓ(d, ϕ) holds. If d ∈ D′ or d ∈ D′′ then the result follows because Abd′ andAbd′′ are both abduction functions. Suppose that d = 〈E, d0〉. Then there is ∆such that PΓ(E,∆) and P∆(d0, ϕ) both hold. By Theorem 3.18 we may assumethat ∆ is finite.

Let E′ ⊆ E be a minimal set such that PΓ(E′, ∆) still holds. Since E′

is finite, let d1, . . . , dn be its elements; by minimality ∆ = ϕ1, . . . , ϕn withPΓ(di, ϕi) holding for i = 1, . . . , n. Then for each i there is Ψi ∈ Abd(di) suchthat PΨi(di, ϕi) holds and Ψi ⊆ Γ. Take Ψ = Ψ1 ∪ . . . Ψn ∈ Abd(d); triviallyΨ ⊆ Γ; furthermore, PΨ(E′, ∆) holds by construction, hence PΨ(E, ∆) alsoholds, and since we assumed P∆(d0, ϕ) we conclude that PΨ(d, ϕ). ¦

We illustrate the abduction function defined above with a simple examplethat also makes clear the need for the complex type of its output.

Example 3.24 Consider the systems P ′ = P(HB) introduced in Example 2.4and P ′′ = P(G∨,⇒,¬), induced by the sequent calculus with connectives ∨, ⇒and ¬ and the rules R⇒, L⇒, R¬ and L¬ from Example 2.6 together with thetwo following rules for ∨.

∆1 → ξ1, ξ2,∆2

∆1 → (ξ1 ∨ ξ2), ∆2R ∨ ∆1, ξ1 → ∆2 ∆1, ξ2 → ∆2

∆1, (ξ1 ∨ ξ2) → ∆2L∨

In P ′, we have the following derivation d1.

1 ξ1 ⇒ (ξ2 ⇒ ξ1) Ax2 ξ1 Hyp3 ξ2 ⇒ ξ1 MP 1,2

According to the definition above, AbdP′(d1) = ξ1.In the same system, we can also build d2 as follows.

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1 (¬ξ2)⇒ ((¬ξ1)⇒ (¬ξ2)) Ax2 (¬ξ2) Hyp3 (¬ξ1)⇒ (¬ξ2) MP 1,24 ((¬ξ1)⇒ (¬ξ2))⇒ (ξ2 ⇒ ξ1) Ax5 ξ2 ⇒ ξ1 MP 4,3

In this case, AbdP ′(d2) = ¬ξ2.Turning now to P ′′, we can build the following derivation d.

1 ξ2 ⇒ ξ1 → ξ1 ∨ (¬ξ2) R∨ 22 ξ2 ⇒ ξ1 → ξ1, (¬ξ2) L⇒ 3,43 → ξ2, ξ1, (¬ξ2) R¬ 54 ξ1 → ξ1, (¬ξ2) Ax5 ξ2 → ξ2, ξ1 Ax

By the definition above, AbdP ′′(d) = ξ2 ⇒ ξ1.Consider now the fibring P = P ′ ] P ′′ and the derivation d∗ = 〈d1, d2, d〉

in this system. The construction in the proof of Theorem 3.23 yields

AbdP(d∗) = ∅, ξ1, ¬ξ2, ξ1,¬ξ2.

It is easy to see that PΓ(d∗, ϕ) holds iff ϕ coincides with ξ1 ∨ (¬ξ2) and Γcontains either ξ1 or ¬ξ2.

Theorem 3.25 Let P ′ and P ′′ be hypothesis-abductible and decidable proofsystems. Then their fibring is a decidable proof system.

Proof: Assume that P ′ and P ′′ are decidable proof systems and let Abd′ andAbd′′ be abduction functions for them. Let Abd be an abduction function forP = P ′ ] P ′′ (in particular, Abd may be the function defined in the proof ofTheorem 3.23). The following recursive algorithm allows us to decide whetherPΓ(d, ϕ) holds.

1. If d ∈ D′ and Pτ ′(Γ)(d, τ ′(ϕ)) holds, then output 1.

2. If d ∈ D′′ and Pτ ′′(Γ)(d, τ ′′(ϕ)) holds, then output 1.

3. If d is not of the form 〈E, d0〉 then output 0.

4. Let d = 〈E, d0〉 and let Abdd = Ψ1, . . . ,Ψn.5. For i = 1, . . . , n do

(a) If PΨi(d0, ϕ) holds and PΓ(E, Ψi) holds, output 1.

(b) Otherwise, increment i.

6. Return 0.

Termination of the algorithm follows from the fact that the abduction functiononly returns a finite number of sets and that P ′, P ′′ and the recursive call to Palways terminate.

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For correctness, suppose first that PΓ(d, ϕ) holds. If d ∈ D′ or d ∈ D′′,then either the first or the second steps will return answer 1. Otherwise, letd = 〈E, d0〉 and suppose that PΓ(E,Ψ) and PΨ(d, ϕ) hold. By definition of ab-duction, P∆(d, ϕ) also holds for some ∆ ∈ Abd(d) with ∆ ⊆ Ψ; this guaranteesthat ∆ will be one of the Ψis in the cycle of the previous algorithm and that thecheck in the first step of the cycle will succeed (using the induction hypothesis).Hence the algorithm always returns 1 in this case.

Conversely, suppost that the algorithm returns 1 to the question PΓ(d, ϕ).If d ∈ D′ or d ∈ D′′, then one of the first two steps must have been used, soPΓ(d, ϕ) holds. Otherwise, d must be of the form 〈E, d0〉 and there is someΨi ∈ Abd(d0) such that PΨi(d0, ϕ) and PΓ(E, Ψi) both hold; but this impliesthat PΓ(d, ϕ) holds. ¦

For illustration, we show how this algorithm works in the previous example.

Example 3.26 Consider again the proof system P presented in the previousexample. For simplicity we omit the translation steps (which would change onlyvariable indices, irrelevant since proof systems are closed under renaming substi-tutions). To decide whether PΓ(d∗, ξ1∨(¬ξ2)) holds, the previous algorithm willfirst compute AbdP′′(d) = (ξ2⇒ ξ1), then check whether P(ξ2⇒ξ1)(d, ξ1 ∨(¬ξ2)) holds (it does), and finally check whether PΓ(d1, d2, (ξ2 ⇒ ξ1)) holds.

We now return briefly to the question of whether recursiveness of the setof theorems is preserved by fibring. Notice that, in general, a derivation withno hypotheses in the fibring of two proof systems requires derivations withhypotheses in the original systems, therefore an affirmative answer is unlikely.Even if the systems are finitary and some version of the Deduction Theoremis available, some form of abduction will be needed to find the intermediaryhypotheses, and since (unlike in the previous case) no derivation is availableto start with, it is not at all obvious how such an abduction mechanism wouldwork. For this reason, we conjecture that recursiveness of theoremhood is notpreserved by fibring, and do not address that question further in the presentpaper.

3.5 Proof systems vs consequence systems

The subsection is dedicated to the investigation of the relationship betweenproof systems and consequence systems. We start by discussing the generationof a consequence system out of a proof system.

Proposition 3.27 A proof system P = 〈C, D, , P 〉 induces a consequencesystem C(P) = 〈C,`〉 where Γ` = ϕ ∈ L(C) : PΓ(D, ϕ).

Proof: (i) Extensivity. Follows directly from the right reflexivity of PΓ. (ii) Mo-notonicity. Suppose that Γ1 ⊆ Γ2 and that ϕ ∈ Γ1. Then PΓ1(D, ϕ) holds. Bythe monotonicity of P we have PΓ1(D, ϕ) ≤ PΓ2(D,ϕ) hence PΓ2(D, ϕ) holdsand so ϕ ∈ Γ`2 . (iii) Idempotence. Suppose that ϕ ∈ (Γ`)`. Then there is d ∈ Dsuch that PΓ`(d, ϕ). On the other hand, there is E ⊆ D such that PΓ(E, Γ`).

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Hence by compositionality in P we have PΓ(E d, ϕ) and so ϕ ∈ Γ`. (iv) Clo-sure for renaming substitution. Assume that ρ is a renaming substitution andϕ ∈ Γ`. Then PΓ(D, ϕ) holds and so, by variable exchange in P, Pρ(Γ)(D, ρ(ϕ))holds and so ρ(ϕ) ∈ ρ(Γ)`. ¦

We will now investigate how properties of the proof system are propagatedto the induced consequence system. It is very simple (similar to closure forrenaming substitution in the proof of the result above) to prove the followingresult:

Proposition 3.28 If P is closed for substitution, then so is C(P).

It is worthwhile to detail compactness, effectiveness and strong effectiveness.

Proposition 3.29 If P is compact, then so is C(P).

Proof: Suppose that P is compact and ϕ ∈ Γ`. Then PΓ(D, ϕ) holds and soby, compactness of P, there is Φ ⊆ Γ finite such that PΦ(D, ϕ) also holds andso ϕ ∈ Φ`. ¦

The following result is very easy to prove.

Proposition 3.30 If P is an effective proof system, then C(P) is semi-decidable.

Proof: Assume that P is an effective proof system and Γ ⊆ L(C) is a recursiveset. Then ϕ ∈ Γ` iff there is d ∈ D such that PΓ(d, ϕ) where PΓ is a recursiverelation. By the projection theorem, Γ` is a recursively enumerable set. ¦

Proposition 3.31 The consequence system C(P) is strongly semi-decidablewhen P is a strongly effective proof system.

Proof: Assume that P is a strongly effective proof system. Let Γ ⊆ L(C) be arecursively enumerable set. Then PΓ is recursively enumerable, so either PΓ = ∅or there is a partial recursive function f : N→ D×L(C) such that f(N) = PΓ.If PΓ = ∅ then Γ` = ∅, which is a recursively enumerable set (in this case Γitself must be empty). Otherwise, define f∗ : N→ L(C) such that f∗(n) is thesecond component of f(n). Clearly f∗ is recursive. Furthermore, ϕ ∈ Γ` iffPΓ(d, ϕ) holds for some d ∈ D iff f(n) = 〈d, ϕ〉 for some n iff f∗(n) = ϕ. Hencethe co-domain of f∗ is precisely Γ` and Γ` is recursively enumerable. ¦

Also of interest is the relationship between a calculus and the proof systemit induces. We do a parametric proof of the following result.

Proposition 3.32 Let Calc be a (Hilbert, sequent, tableau) calculus. ThenC(P(Calc)) = C(Calc).

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Proof: Since both C(P(Calc)) and C(Calc) share the same signature C, all oneneeds to show is that the closure of a set Γ ⊆ L(C) is the same in both cases.Let C(P(Calc)) = 〈C,`1〉 and C(Calc) = 〈C,`2〉. Then ϕ ∈ Γ`1 iff PΓ(D, ϕ)holds in P(H) iff there is a Calc-derivation of ϕ from Γ in D iff ϕ ∈ Γ`2 . ¦

The following result indicates that relationships between proof systems arepreserved by the induced consequence systems.

Proposition 3.33 Let P and P ′ be proof systems such that P ≤ P ′. ThenC(P ) ≤ C(P ′).

Proof: Let P ≤ P ′ and Γ ⊆ L(C). Then C ⊆ C ′; suppose that ϕ ∈ Γ`. ThenPΓ(D, ϕ) holds, whence P ′

Γ(D′, ϕ) also holds since P ≤ P ′ and so ϕ ∈ Γ`′ . ¦

As a special case we conclude C(P ′) ≤ C(P ′ ] P ′′) and C(P ′′) ≤ C(P ′ ] P ′′).Now we show how to generate a proof system out of a consequence system.

Proposition 3.34 A consequence system C induces a proof system P(C) withthe same signature as follows: D = ∗; E ∗ = ∗; PΓ(∗, ϕ) holds iff ϕ ∈ Γ`.

Proof: (i) Right reflexivity. Since ` is extensive, Γ ⊆ Γ` for every Γ ⊆ L(C), soPΓ(D, Γ) holds. (ii) Monotonicity. Assume that Γ1 ⊆ Γ2 and PΓ1(D, ϕ) holds.Then ϕ ∈ Γ`1 ; by monotonicity of C, Γ`1 ⊆ Γ`2 , hence ϕ ∈ Γ`2 , and so PΓ2(D,ϕ).(iii) Idempotence. Suppose that PΓ(E,Ψ) and PΨ(d, ϕ) hold. Then Ψ ⊆ Γ`

and ϕ ∈ Ψ`, hence, by monotonicity of C, ϕ ∈ (Γ`)` and so, by idempotenceof `, ϕ ∈ Γ`. Therefore PΓ(E d, ϕ) holds. (iv) Variable exchange. Assumethat ρ is a renaming substitution and that that PΓ(D, ϕ) holds. Then ϕ ∈ Γ`,hence ρ(ϕ) ∈ ρ(Γ)` and so Pρ(Γ)(D, ρ(ϕ)). ¦

We can show easily that the induced proof system is closed for substitu-tion and compact whenever the consequence system has the same properties.A proof system P can be compared with the proof system generated by theconsequence system induced by P as the following result states.

Proposition 3.35 For any proof system P, P ≤ P(C(P)).

Proof: Straightforward. Since in both constructions the signature does notchange, all that is left to show is that, if PΓ(D,ϕ) holds in P, then PΓ(∗, ϕ)holds in P(C(P)). Assume that PΓ(D, ϕ) holds in P. Then ϕ ∈ Γ` and soPΓ(∗, ϕ) holds in P(C(P)). ¦

The opposite relation also holds: for every consequence system: C = C(P(C)).Finally, we relate the consequence system induced by the fibring of proof

systems with the fibring of the consequence systems induced by the proof sys-tems.

Proposition 3.36 The fibring of proof systems has the following property.

C(P ′ ] P ′′) = C(P ′) ] C(P ′′)

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Proof: The signature of both C(P ′ ] P ′′) and C(P ′) ] C(P ′′) is C = C ′ ∪ C ′′.Denoting C(P ′ ]P ′′) by 〈C,`a〉 and C(P ′)] C(P ′′) by 〈C,`b〉, all that is left toshow is that Γ`a = Γ`b for all Γ ⊆ L(C).(i) We start by showing that Γ`a ⊆ Γ`b . Suppose ϕ ∈ Γ`a . Then PΓ(D, ϕ)holds, hence PΓ(d, ϕ) holds for some d ∈ D. We prove that ϕ ∈ Γ`b by inductionon d. (a) If d is d′ ∈ D′, then P ′

τ ′(Γ)(d′, τ ′(ϕ)), hence τ ′(ϕ) ∈ τ ′(Γ)`′ and

therefore ϕ ∈ Γ`b by definition of fibring of consequence systems. The casewhere d is d′′ ∈ D′′ is analogous. (b) If d is 〈E, d′′′〉 with E ∪ d′′′ ⊆ D,then there is a set Ψ such that PΓ(E,Ψ) and PΨ(d′′′, ϕ) both hold, that is,Ψ ⊆ Γ`a and ϕ ∈ Ψ`a . By induction hypothesis, Ψ ⊆ Γ`b and ϕ ∈ Ψ`b and, byidempotence of `b, it follows that ϕ ∈ (Γ`b)`b ⊆ Γ`b .(ii) Now we show that Γ`b ⊆ Γ`a . Suppose now that ϕ ∈ Γ`b . Then ϕ ∈ Γ`β

for some ordinal β in the fixed point construction of Proposition 2.13. We provethat ϕ ∈ Γ`a by induction on β. (a) β = 0. Straightforward, since Γ ⊆ Γ`a .(b) If ϕ ∈ Γ`β+1 , then either ϕ ∈ τ ′−1(τ ′(Γ`β )`′) or ϕ ∈ τ ′′−1(τ ′′(Γ`β )`′′);both cases are similar, so assume the first one holds. By induction hypothesisΓ`β ⊆ Γ`a , so PΓ(D, Γ`β ) holds. Also, from ϕ ∈ τ ′−1(τ ′(Γ`β )`′) we concludethat τ ′(ϕ) ∈ τ ′(Γ`β )`′ , so P ′

τ ′(Γ`β )(d′, τ ′(ϕ)) holds for some d′ ∈ D′. Therefore,

PΓ`β (d′, ϕ) also holds and hence PΓ(Dd′, ϕ) holds, which means that ϕ ∈ Γ`a .

(c) β is a limit ordinal: straightforward. ¦

4 Conclusions

In this paper we addressed the problem of heterogenous fibring of logics. In thefirst place, we studied the well-known notion of consequence system, showed howseveral kinds of presentations of logics define consequence systems and definedfibring of consequence systems. As a consequence, we showed how we couldcombine a logic presented syntactically with another presented semantically.We showed that this combination is conservative assuming the original logicsare closed under substitution, as well as several results on semi-decidability.

However, this solution is unsatisfactory because no trace is kept of proofsthat may exist in the original calculi. For this reason, we introduced the notionof abstract proof system, which intends to abstract the essential properties oflogics presented syntactically via some notion of derivation. In particular, thiscovers Hilbert calculi, sequent calculi and tableau calculi. We showed how todefine fibring in this context, and gave some examples illustrating this con-struction and its advantages over the analogous one obtained by regarding thelogics as consequence systems. Finally we defined the property of hypothesis-abduction for an abstract proof system and showed how it could be used toprove preservation of semi-decidability by fibring of proof systems.

Finally we showed how every proof system can be seen as a consequencesystem and vice-versa, so that even semantically presented logics can be seento induce abstract proof systems (albeit not-so-interesting ones), and showedthat fibring commutes with these views for all the concrete calculi considered.

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4.1 Future Work

There are some issues raised in this paper that were not fully explored for lackof space.

First-order case Throughout this paper we only considered propositionalsignatures for two reasons: they are clearly easier to combine in a natural way,and they suffice for many practical applications. Generalizing the notion ofproof system to encompass first- (and higher-) order logics presents differentchallenges already at that level. Furthermore, since derivations in first-orderlogic already require the use of rules with provisos, the mechanism of combina-tion itself has to be revised.

Theoremhood As explained at the end of Section 3.4, the question of whetherrecursiveness of theoremhood is preserved by fibring is far from trivial. Decidingthis question will likely require stating the Deduction Theorem in general forproof systems and analyzing how it behaves through fibring, as well as somemore work on abduction of hypotheses.

Structural properties of derivations Since we work with proof systems in-duced from specific calculi, it would be worthwhile to explore properties ofderivations that can be abstracted to families of proof systems, and whetherthey are preserved through fibring. Examples of such properties are the rela-tionship of the size of a formula to the size of its derivation, or invertibility ofrules in the case of derivations produced from rules. Another example, withobvious implications in the question discussed in the previous paragraph, iswhether a derivation can be generated simply from the structure of a givenformula.

Complexity The abduction algorithm for the fibring of two hypothesis-abductible proof systems is very simple and, as such, terribly inefficient. Itwould be interesting to examine how efficient it can be made and whether goodbounds on the time complexity of abduction in the fibring can be proved, basedon bounds on the time complexity of the abduction in the components. At thevery least, it would be good to know that fibring preserves polynomial-timeabduction.

Acknowledgments

The work was supported by Fundacao para a Ciencia e Tecnologia (FCT) andEU FEDER, namely via the Projects FibLog POCTI / MAT / 37239 / 2001and Quantlog POCI / MAT / 55796 / 2004 at CLC (Center for Logic andComputation) of IST. Luıs Cruz-Filipe was also supported by FCT throughpostdoc grant SFRH / BPD / 16372 / 2004 and the European Project IST-2005-16004. The authors want to thank the anonymous referee for many usefulsuggestions.

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